diff --git a/pandas/core/arrays/_arrow_string_mixins.py b/pandas/core/arrays/_arrow_string_mixins.py
index a39668faf779e..e136b4f92031d 100644
--- a/pandas/core/arrays/_arrow_string_mixins.py
+++ b/pandas/core/arrays/_arrow_string_mixins.py
@@ -10,6 +10,7 @@
 
 import numpy as np
 
+from pandas._libs import lib
 from pandas.compat import (
     pa_version_under10p1,
     pa_version_under11p0,
@@ -17,8 +18,6 @@
     pa_version_under17p0,
 )
 
-from pandas.core.dtypes.missing import isna
-
 if not pa_version_under10p1:
     import pyarrow as pa
     import pyarrow.compute as pc
@@ -38,7 +37,7 @@ class ArrowStringArrayMixin:
     def __init__(self, *args, **kwargs) -> None:
         raise NotImplementedError
 
-    def _convert_bool_result(self, result):
+    def _convert_bool_result(self, result, na=lib.no_default, method_name=None):
         # Convert a bool-dtype result to the appropriate result type
         raise NotImplementedError
 
@@ -212,7 +211,9 @@ def _str_removesuffix(self, suffix: str):
         result = pc.if_else(ends_with, removed, self._pa_array)
         return type(self)(result)
 
-    def _str_startswith(self, pat: str | tuple[str, ...], na: Scalar | None = None):
+    def _str_startswith(
+        self, pat: str | tuple[str, ...], na: Scalar | lib.NoDefault = lib.no_default
+    ):
         if isinstance(pat, str):
             result = pc.starts_with(self._pa_array, pattern=pat)
         else:
@@ -225,11 +226,11 @@ def _str_startswith(self, pat: str | tuple[str, ...], na: Scalar | None = None):
 
                 for p in pat[1:]:
                     result = pc.or_(result, pc.starts_with(self._pa_array, pattern=p))
-        if not isna(na):  # pyright: ignore [reportGeneralTypeIssues]
-            result = result.fill_null(na)
-        return self._convert_bool_result(result)
+        return self._convert_bool_result(result, na=na, method_name="startswith")
 
-    def _str_endswith(self, pat: str | tuple[str, ...], na: Scalar | None = None):
+    def _str_endswith(
+        self, pat: str | tuple[str, ...], na: Scalar | lib.NoDefault = lib.no_default
+    ):
         if isinstance(pat, str):
             result = pc.ends_with(self._pa_array, pattern=pat)
         else:
@@ -242,9 +243,7 @@ def _str_endswith(self, pat: str | tuple[str, ...], na: Scalar | None = None):
 
                 for p in pat[1:]:
                     result = pc.or_(result, pc.ends_with(self._pa_array, pattern=p))
-        if not isna(na):  # pyright: ignore [reportGeneralTypeIssues]
-            result = result.fill_null(na)
-        return self._convert_bool_result(result)
+        return self._convert_bool_result(result, na=na, method_name="endswith")
 
     def _str_isalnum(self):
         result = pc.utf8_is_alnum(self._pa_array)
@@ -283,7 +282,12 @@ def _str_isupper(self):
         return self._convert_bool_result(result)
 
     def _str_contains(
-        self, pat, case: bool = True, flags: int = 0, na=None, regex: bool = True
+        self,
+        pat,
+        case: bool = True,
+        flags: int = 0,
+        na: Scalar | lib.NoDefault = lib.no_default,
+        regex: bool = True,
     ):
         if flags:
             raise NotImplementedError(f"contains not implemented with {flags=}")
@@ -293,19 +297,25 @@ def _str_contains(
         else:
             pa_contains = pc.match_substring
         result = pa_contains(self._pa_array, pat, ignore_case=not case)
-        if not isna(na):  # pyright: ignore [reportGeneralTypeIssues]
-            result = result.fill_null(na)
-        return self._convert_bool_result(result)
+        return self._convert_bool_result(result, na=na, method_name="contains")
 
     def _str_match(
-        self, pat: str, case: bool = True, flags: int = 0, na: Scalar | None = None
+        self,
+        pat: str,
+        case: bool = True,
+        flags: int = 0,
+        na: Scalar | lib.NoDefault = lib.no_default,
     ):
         if not pat.startswith("^"):
             pat = f"^{pat}"
         return self._str_contains(pat, case, flags, na, regex=True)
 
     def _str_fullmatch(
-        self, pat, case: bool = True, flags: int = 0, na: Scalar | None = None
+        self,
+        pat,
+        case: bool = True,
+        flags: int = 0,
+        na: Scalar | lib.NoDefault = lib.no_default,
     ):
         if not pat.endswith("$") or pat.endswith("\\$"):
             pat = f"{pat}$"
diff --git a/pandas/core/arrays/arrow/array.py b/pandas/core/arrays/arrow/array.py
index e0ccbd6fdc5fd..f3d7a3cc6d694 100644
--- a/pandas/core/arrays/arrow/array.py
+++ b/pandas/core/arrays/arrow/array.py
@@ -2285,7 +2285,11 @@ def _apply_elementwise(self, func: Callable) -> list[list[Any]]:
             for chunk in self._pa_array.iterchunks()
         ]
 
-    def _convert_bool_result(self, result):
+    def _convert_bool_result(self, result, na=lib.no_default, method_name=None):
+        if na is not lib.no_default and not isna(
+            na
+        ):  # pyright: ignore [reportGeneralTypeIssues]
+            result = result.fill_null(na)
         return type(self)(result)
 
     def _convert_int_result(self, result):
diff --git a/pandas/core/arrays/categorical.py b/pandas/core/arrays/categorical.py
index 97004474648b2..366253a923f6c 100644
--- a/pandas/core/arrays/categorical.py
+++ b/pandas/core/arrays/categorical.py
@@ -2675,16 +2675,28 @@ def _replace(self, *, to_replace, value, inplace: bool = False):
     # ------------------------------------------------------------------------
     # String methods interface
     def _str_map(
-        self, f, na_value=np.nan, dtype=np.dtype("object"), convert: bool = True
+        self, f, na_value=lib.no_default, dtype=np.dtype("object"), convert: bool = True
     ):
         # Optimization to apply the callable `f` to the categories once
         # and rebuild the result by `take`ing from the result with the codes.
         # Returns the same type as the object-dtype implementation though.
-        from pandas.core.arrays import NumpyExtensionArray
-
         categories = self.categories
         codes = self.codes
-        result = NumpyExtensionArray(categories.to_numpy())._str_map(f, na_value, dtype)
+        if categories.dtype == "string":
+            result = categories.array._str_map(f, na_value, dtype)  # type: ignore[attr-defined]
+            if (
+                categories.dtype.na_value is np.nan  # type: ignore[union-attr]
+                and is_bool_dtype(dtype)
+                and (na_value is lib.no_default or isna(na_value))
+            ):
+                # NaN propagates as False for functions with boolean return type
+                na_value = False
+        else:
+            from pandas.core.arrays import NumpyExtensionArray
+
+            result = NumpyExtensionArray(categories.to_numpy())._str_map(
+                f, na_value, dtype
+            )
         return take_nd(result, codes, fill_value=na_value)
 
     def _str_get_dummies(self, sep: str = "|"):
diff --git a/pandas/core/arrays/string_.py b/pandas/core/arrays/string_.py
index 0b0fffcb928a3..5b69344bac0c8 100644
--- a/pandas/core/arrays/string_.py
+++ b/pandas/core/arrays/string_.py
@@ -377,7 +377,11 @@ def _from_scalars(cls, scalars, dtype: DtypeObj) -> Self:
         return cls._from_sequence(scalars, dtype=dtype)
 
     def _str_map(
-        self, f, na_value=None, dtype: Dtype | None = None, convert: bool = True
+        self,
+        f,
+        na_value=lib.no_default,
+        dtype: Dtype | None = None,
+        convert: bool = True,
     ):
         if self.dtype.na_value is np.nan:
             return self._str_map_nan_semantics(
@@ -388,7 +392,7 @@ def _str_map(
 
         if dtype is None:
             dtype = self.dtype
-        if na_value is None:
+        if na_value is lib.no_default:
             na_value = self.dtype.na_value
 
         mask = isna(self)
@@ -458,12 +462,20 @@ def _str_map_str_or_object(
             return lib.map_infer_mask(arr, f, mask.view("uint8"))
 
     def _str_map_nan_semantics(
-        self, f, na_value=None, dtype: Dtype | None = None, convert: bool = True
+        self,
+        f,
+        na_value=lib.no_default,
+        dtype: Dtype | None = None,
+        convert: bool = True,
     ):
         if dtype is None:
             dtype = self.dtype
-        if na_value is None:
-            na_value = self.dtype.na_value
+        if na_value is lib.no_default:
+            if is_bool_dtype(dtype):
+                # NaN propagates as False
+                na_value = False
+            else:
+                na_value = self.dtype.na_value
 
         mask = isna(self)
         arr = np.asarray(self)
@@ -474,7 +486,8 @@ def _str_map_nan_semantics(
                 if is_integer_dtype(dtype):
                     na_value = 0
                 else:
-                    na_value = True
+                    # NaN propagates as False
+                    na_value = False
 
             result = lib.map_infer_mask(
                 arr,
@@ -484,15 +497,13 @@ def _str_map_nan_semantics(
                 na_value=na_value,
                 dtype=np.dtype(cast(type, dtype)),
             )
-            if na_value_is_na and mask.any():
+            if na_value_is_na and is_integer_dtype(dtype) and mask.any():
                 # TODO: we could alternatively do this check before map_infer_mask
                 #  and adjust the dtype/na_value we pass there. Which is more
                 #  performant?
-                if is_integer_dtype(dtype):
-                    result = result.astype("float64")
-                else:
-                    result = result.astype("object")
+                result = result.astype("float64")
                 result[mask] = np.nan
+
             return result
 
         else:
diff --git a/pandas/core/arrays/string_arrow.py b/pandas/core/arrays/string_arrow.py
index 8dcf7643b579e..9389f7cffca9f 100644
--- a/pandas/core/arrays/string_arrow.py
+++ b/pandas/core/arrays/string_arrow.py
@@ -211,9 +211,29 @@ def insert(self, loc: int, item) -> ArrowStringArray:
             raise TypeError("Scalar must be NA or str")
         return super().insert(loc, item)
 
-    def _convert_bool_result(self, values):
+    def _convert_bool_result(self, values, na=lib.no_default, method_name=None):
+        if na is not lib.no_default and not isna(na) and not isinstance(na, bool):
+            # GH#59561
+            warnings.warn(
+                f"Allowing a non-bool 'na' in obj.str.{method_name} is deprecated "
+                "and will raise in a future version.",
+                FutureWarning,
+                stacklevel=find_stack_level(),
+            )
+            na = bool(na)
+
         if self.dtype.na_value is np.nan:
-            return ArrowExtensionArray(values).to_numpy(na_value=np.nan)
+            if na is lib.no_default or isna(na):
+                # NaN propagates as False
+                values = values.fill_null(False)
+            else:
+                values = values.fill_null(na)
+            return values.to_numpy()
+        else:
+            if na is not lib.no_default and not isna(
+                na
+            ):  # pyright: ignore [reportGeneralTypeIssues]
+                values = values.fill_null(na)
         return BooleanDtype().__from_arrow__(values)
 
     def _maybe_convert_setitem_value(self, value):
@@ -309,22 +329,16 @@ def _data(self):
     _str_slice = ArrowStringArrayMixin._str_slice
 
     def _str_contains(
-        self, pat, case: bool = True, flags: int = 0, na=np.nan, regex: bool = True
+        self,
+        pat,
+        case: bool = True,
+        flags: int = 0,
+        na=lib.no_default,
+        regex: bool = True,
     ):
         if flags:
             return super()._str_contains(pat, case, flags, na, regex)
 
-        if not isna(na):
-            if not isinstance(na, bool):
-                # GH#59561
-                warnings.warn(
-                    "Allowing a non-bool 'na' in obj.str.contains is deprecated "
-                    "and will raise in a future version.",
-                    FutureWarning,
-                    stacklevel=find_stack_level(),
-                )
-                na = bool(na)
-
         return ArrowStringArrayMixin._str_contains(self, pat, case, flags, na, regex)
 
     def _str_replace(
diff --git a/pandas/core/strings/accessor.py b/pandas/core/strings/accessor.py
index da10a12d02ae4..563dce3008480 100644
--- a/pandas/core/strings/accessor.py
+++ b/pandas/core/strings/accessor.py
@@ -1199,7 +1199,12 @@ def join(self, sep: str):
 
     @forbid_nonstring_types(["bytes"])
     def contains(
-        self, pat, case: bool = True, flags: int = 0, na=None, regex: bool = True
+        self,
+        pat,
+        case: bool = True,
+        flags: int = 0,
+        na=lib.no_default,
+        regex: bool = True,
     ):
         r"""
         Test if pattern or regex is contained within a string of a Series or Index.
@@ -1217,8 +1222,9 @@ def contains(
             Flags to pass through to the re module, e.g. re.IGNORECASE.
         na : scalar, optional
             Fill value for missing values. The default depends on dtype of the
-            array. For object-dtype, ``numpy.nan`` is used. For ``StringDtype``,
-            ``pandas.NA`` is used.
+            array. For object-dtype, ``numpy.nan`` is used. For the nullable
+            ``StringDtype``, ``pandas.NA`` is used. For the ``"str"`` dtype,
+            ``False`` is used.
         regex : bool, default True
             If True, assumes the pat is a regular expression.
 
@@ -1336,7 +1342,7 @@ def contains(
         return self._wrap_result(result, fill_value=na, returns_string=False)
 
     @forbid_nonstring_types(["bytes"])
-    def match(self, pat: str, case: bool = True, flags: int = 0, na=None):
+    def match(self, pat: str, case: bool = True, flags: int = 0, na=lib.no_default):
         """
         Determine if each string starts with a match of a regular expression.
 
@@ -1350,8 +1356,9 @@ def match(self, pat: str, case: bool = True, flags: int = 0, na=None):
             Regex module flags, e.g. re.IGNORECASE.
         na : scalar, optional
             Fill value for missing values. The default depends on dtype of the
-            array. For object-dtype, ``numpy.nan`` is used. For ``StringDtype``,
-            ``pandas.NA`` is used.
+            array. For object-dtype, ``numpy.nan`` is used. For the nullable
+            ``StringDtype``, ``pandas.NA`` is used. For the ``"str"`` dtype,
+            ``False`` is used.
 
         Returns
         -------
@@ -1377,7 +1384,7 @@ def match(self, pat: str, case: bool = True, flags: int = 0, na=None):
         return self._wrap_result(result, fill_value=na, returns_string=False)
 
     @forbid_nonstring_types(["bytes"])
-    def fullmatch(self, pat, case: bool = True, flags: int = 0, na=None):
+    def fullmatch(self, pat, case: bool = True, flags: int = 0, na=lib.no_default):
         """
         Determine if each string entirely matches a regular expression.
 
@@ -1391,8 +1398,9 @@ def fullmatch(self, pat, case: bool = True, flags: int = 0, na=None):
             Regex module flags, e.g. re.IGNORECASE.
         na : scalar, optional
             Fill value for missing values. The default depends on dtype of the
-            array. For object-dtype, ``numpy.nan`` is used. For ``StringDtype``,
-            ``pandas.NA`` is used.
+            array. For object-dtype, ``numpy.nan`` is used. For the nullable
+            ``StringDtype``, ``pandas.NA`` is used. For the ``"str"`` dtype,
+            ``False`` is used.
 
         Returns
         -------
@@ -2415,7 +2423,7 @@ def count(self, pat, flags: int = 0):
 
     @forbid_nonstring_types(["bytes"])
     def startswith(
-        self, pat: str | tuple[str, ...], na: Scalar | None = None
+        self, pat: str | tuple[str, ...], na: Scalar | lib.NoDefault = lib.no_default
     ) -> Series | Index:
         """
         Test if the start of each string element matches a pattern.
@@ -2427,10 +2435,11 @@ def startswith(
         pat : str or tuple[str, ...]
             Character sequence or tuple of strings. Regular expressions are not
             accepted.
-        na : object, default NaN
+        na : scalar, optional
             Object shown if element tested is not a string. The default depends
             on dtype of the array. For object-dtype, ``numpy.nan`` is used.
-            For ``StringDtype``, ``pandas.NA`` is used.
+            For the nullable ``StringDtype``, ``pandas.NA`` is used.
+            For the ``"str"`` dtype, ``False`` is used.
 
         Returns
         -------
@@ -2485,7 +2494,7 @@ def startswith(
 
     @forbid_nonstring_types(["bytes"])
     def endswith(
-        self, pat: str | tuple[str, ...], na: Scalar | None = None
+        self, pat: str | tuple[str, ...], na: Scalar | lib.NoDefault = lib.no_default
     ) -> Series | Index:
         """
         Test if the end of each string element matches a pattern.
@@ -2497,10 +2506,11 @@ def endswith(
         pat : str or tuple[str, ...]
             Character sequence or tuple of strings. Regular expressions are not
             accepted.
-        na : object, default NaN
+        na : scalar, optional
             Object shown if element tested is not a string. The default depends
             on dtype of the array. For object-dtype, ``numpy.nan`` is used.
-            For ``StringDtype``, ``pandas.NA`` is used.
+            For the nullable ``StringDtype``, ``pandas.NA`` is used.
+            For the ``"str"`` dtype, ``False`` is used.
 
         Returns
         -------
diff --git a/pandas/core/strings/base.py b/pandas/core/strings/base.py
index 96b0352666b41..316c86d152db3 100644
--- a/pandas/core/strings/base.py
+++ b/pandas/core/strings/base.py
@@ -7,7 +7,7 @@
     Literal,
 )
 
-import numpy as np
+from pandas._libs import lib
 
 if TYPE_CHECKING:
     from collections.abc import Sequence
@@ -85,7 +85,11 @@ def _str_repeat(self, repeats: int | Sequence[int]):
 
     @abc.abstractmethod
     def _str_match(
-        self, pat: str, case: bool = True, flags: int = 0, na: Scalar = np.nan
+        self,
+        pat: str,
+        case: bool = True,
+        flags: int = 0,
+        na: Scalar | lib.NoDefault = lib.no_default,
     ):
         pass
 
@@ -95,7 +99,7 @@ def _str_fullmatch(
         pat: str | re.Pattern,
         case: bool = True,
         flags: int = 0,
-        na: Scalar = np.nan,
+        na: Scalar | lib.NoDefault = lib.no_default,
     ):
         pass
 
diff --git a/pandas/core/strings/object_array.py b/pandas/core/strings/object_array.py
index f376c239a0ce0..e82c6c20e86d9 100644
--- a/pandas/core/strings/object_array.py
+++ b/pandas/core/strings/object_array.py
@@ -44,7 +44,11 @@ def __len__(self) -> int:
         raise NotImplementedError
 
     def _str_map(
-        self, f, na_value=None, dtype: NpDtype | None = None, convert: bool = True
+        self,
+        f,
+        na_value=lib.no_default,
+        dtype: NpDtype | None = None,
+        convert: bool = True,
     ):
         """
         Map a callable over valid elements of the array.
@@ -65,7 +69,7 @@ def _str_map(
         """
         if dtype is None:
             dtype = np.dtype("object")
-        if na_value is None:
+        if na_value is lib.no_default:
             na_value = self.dtype.na_value  # type: ignore[attr-defined]
 
         if not len(self):
@@ -127,7 +131,12 @@ def _str_pad(
         return self._str_map(f)
 
     def _str_contains(
-        self, pat, case: bool = True, flags: int = 0, na=np.nan, regex: bool = True
+        self,
+        pat,
+        case: bool = True,
+        flags: int = 0,
+        na=lib.no_default,
+        regex: bool = True,
     ):
         if regex:
             if not case:
@@ -142,7 +151,7 @@ def _str_contains(
             else:
                 upper_pat = pat.upper()
                 f = lambda x: upper_pat in x.upper()
-        if not isna(na) and not isinstance(na, bool):
+        if na is not lib.no_default and not isna(na) and not isinstance(na, bool):
             # GH#59561
             warnings.warn(
                 "Allowing a non-bool 'na' in obj.str.contains is deprecated "
@@ -152,9 +161,9 @@ def _str_contains(
             )
         return self._str_map(f, na, dtype=np.dtype("bool"))
 
-    def _str_startswith(self, pat, na=None):
+    def _str_startswith(self, pat, na=lib.no_default):
         f = lambda x: x.startswith(pat)
-        if not isna(na) and not isinstance(na, bool):
+        if na is not lib.no_default and not isna(na) and not isinstance(na, bool):
             # GH#59561
             warnings.warn(
                 "Allowing a non-bool 'na' in obj.str.startswith is deprecated "
@@ -164,9 +173,9 @@ def _str_startswith(self, pat, na=None):
             )
         return self._str_map(f, na_value=na, dtype=np.dtype(bool))
 
-    def _str_endswith(self, pat, na=None):
+    def _str_endswith(self, pat, na=lib.no_default):
         f = lambda x: x.endswith(pat)
-        if not isna(na) and not isinstance(na, bool):
+        if na is not lib.no_default and not isna(na) and not isinstance(na, bool):
             # GH#59561
             warnings.warn(
                 "Allowing a non-bool 'na' in obj.str.endswith is deprecated "
@@ -235,7 +244,11 @@ def rep(x, r):
             return result
 
     def _str_match(
-        self, pat: str, case: bool = True, flags: int = 0, na: Scalar | None = None
+        self,
+        pat: str,
+        case: bool = True,
+        flags: int = 0,
+        na: Scalar | lib.NoDefault = lib.no_default,
     ):
         if not case:
             flags |= re.IGNORECASE
@@ -250,7 +263,7 @@ def _str_fullmatch(
         pat: str | re.Pattern,
         case: bool = True,
         flags: int = 0,
-        na: Scalar | None = None,
+        na: Scalar | lib.NoDefault = lib.no_default,
     ):
         if not case:
             flags |= re.IGNORECASE
diff --git a/pandas/tests/strings/test_api.py b/pandas/tests/strings/test_api.py
index 31e005466af7b..8987fc36656c5 100644
--- a/pandas/tests/strings/test_api.py
+++ b/pandas/tests/strings/test_api.py
@@ -111,6 +111,7 @@ def test_api_per_method(
     any_allowed_skipna_inferred_dtype,
     any_string_method,
     request,
+    using_infer_string,
 ):
     # this test does not check correctness of the different methods,
     # just that the methods work on the specified (inferred) dtypes,
@@ -149,6 +150,10 @@ def test_api_per_method(
     t = box(values, dtype=dtype)  # explicit dtype to avoid casting
     method = getattr(t.str, method_name)
 
+    if using_infer_string and dtype == "category":
+        string_allowed = method_name not in ["decode"]
+    else:
+        string_allowed = True
     bytes_allowed = method_name in ["decode", "get", "len", "slice"]
     # as of v0.23.4, all methods except 'cat' are very lenient with the
     # allowed data types, just returning NaN for entries that error.
@@ -157,7 +162,8 @@ def test_api_per_method(
     mixed_allowed = method_name not in ["cat"]
 
     allowed_types = (
-        ["string", "unicode", "empty"]
+        ["empty"]
+        + ["string", "unicode"] * string_allowed
         + ["bytes"] * bytes_allowed
         + ["mixed", "mixed-integer"] * mixed_allowed
     )
@@ -171,6 +177,7 @@ def test_api_per_method(
         msg = (
             f"Cannot use .str.{method_name} with values of "
             f"inferred dtype {repr(inferred_dtype)}."
+            "|a bytes-like object is required, not 'str'"
         )
         with pytest.raises(TypeError, match=msg):
             method(*args, **kwargs)
diff --git a/pandas/tests/strings/test_find_replace.py b/pandas/tests/strings/test_find_replace.py
index 2742c5b67e57e..48159c07de6ab 100644
--- a/pandas/tests/strings/test_find_replace.py
+++ b/pandas/tests/strings/test_find_replace.py
@@ -29,20 +29,28 @@ def test_contains(any_string_dtype):
     pat = "mmm[_]+"
 
     result = values.str.contains(pat)
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
-    expected = Series(
-        np.array([False, np.nan, True, True, False], dtype=np.object_),
-        dtype=expected_dtype,
-    )
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected = Series([False, False, True, True, False], dtype=bool)
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        expected = Series(
+            np.array([False, np.nan, True, True, False], dtype=np.object_),
+            dtype=expected_dtype,
+        )
+
     tm.assert_series_equal(result, expected)
 
     result = values.str.contains(pat, regex=False)
-    expected = Series(
-        np.array([False, np.nan, False, False, True], dtype=np.object_),
-        dtype=expected_dtype,
-    )
+    if any_string_dtype == "str":
+        expected = Series([False, False, False, False, True], dtype=bool)
+    else:
+        expected = Series(
+            np.array([False, np.nan, False, False, True], dtype=np.object_),
+            dtype=expected_dtype,
+        )
     tm.assert_series_equal(result, expected)
 
     values = Series(
@@ -79,12 +87,16 @@ def test_contains(any_string_dtype):
     pat = "mmm[_]+"
 
     result = values.str.contains(pat)
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
-    expected = Series(
-        np.array([False, np.nan, True, True], dtype=np.object_), dtype=expected_dtype
-    )
+    if any_string_dtype == "str":
+        expected = Series([False, False, True, True], dtype=bool)
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        expected = Series(
+            np.array([False, np.nan, True, True], dtype=np.object_),
+            dtype=expected_dtype,
+        )
     tm.assert_series_equal(result, expected)
 
     result = values.str.contains(pat, na=False)
@@ -184,39 +196,45 @@ def test_contains_moar(any_string_dtype):
     )
 
     result = s.str.contains("a")
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected_dtype = bool
+        na_value = False
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        na_value = np.nan
     expected = Series(
-        [False, False, False, True, True, False, np.nan, False, False, True],
+        [False, False, False, True, True, False, na_value, False, False, True],
         dtype=expected_dtype,
     )
     tm.assert_series_equal(result, expected)
 
     result = s.str.contains("a", case=False)
     expected = Series(
-        [True, False, False, True, True, False, np.nan, True, False, True],
+        [True, False, False, True, True, False, na_value, True, False, True],
         dtype=expected_dtype,
     )
     tm.assert_series_equal(result, expected)
 
     result = s.str.contains("Aa")
     expected = Series(
-        [False, False, False, True, False, False, np.nan, False, False, False],
+        [False, False, False, True, False, False, na_value, False, False, False],
         dtype=expected_dtype,
     )
     tm.assert_series_equal(result, expected)
 
     result = s.str.contains("ba")
     expected = Series(
-        [False, False, False, True, False, False, np.nan, False, False, False],
+        [False, False, False, True, False, False, na_value, False, False, False],
         dtype=expected_dtype,
     )
     tm.assert_series_equal(result, expected)
 
     result = s.str.contains("ba", case=False)
     expected = Series(
-        [False, False, False, True, True, False, np.nan, True, False, False],
+        [False, False, False, True, True, False, na_value, True, False, False],
         dtype=expected_dtype,
     )
     tm.assert_series_equal(result, expected)
@@ -261,10 +279,14 @@ def test_contains_nan(any_string_dtype):
         tm.assert_series_equal(result, expected)
 
     result = s.str.contains("foo")
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
-    expected = Series([np.nan, np.nan, np.nan], dtype=expected_dtype)
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected = Series([False, False, False], dtype=bool)
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        expected = Series([np.nan, np.nan, np.nan], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
 
@@ -287,9 +309,7 @@ def test_startswith_endswith_validate_na(request, any_string_dtype):
     )
 
     dtype = ser.dtype
-    if (
-        isinstance(dtype, pd.StringDtype) and dtype.storage == "python"
-    ) or dtype == np.dtype("object"):
+    if (isinstance(dtype, pd.StringDtype)) or dtype == np.dtype("object"):
         msg = "Allowing a non-bool 'na' in obj.str.startswith is deprecated"
         with tm.assert_produces_warning(FutureWarning, match=msg):
             ser.str.startswith("kapow", na="baz")
@@ -307,11 +327,12 @@ def test_startswith_endswith_validate_na(request, any_string_dtype):
             ser.str.endswith("kapow", na="baz")
 
 
+@pytest.mark.filterwarnings("ignore:Downcasting object dtype arrays:FutureWarning")
 @pytest.mark.parametrize("pat", ["foo", ("foo", "baz")])
 @pytest.mark.parametrize("dtype", ["object", "category"])
 @pytest.mark.parametrize("null_value", [None, np.nan, pd.NA])
 @pytest.mark.parametrize("na", [True, False])
-def test_startswith(pat, dtype, null_value, na):
+def test_startswith(pat, dtype, null_value, na, using_infer_string):
     # add category dtype parametrizations for GH-36241
     values = Series(
         ["om", null_value, "foo_nom", "nom", "bar_foo", null_value, "foo"],
@@ -325,6 +346,8 @@ def test_startswith(pat, dtype, null_value, na):
         exp = exp.fillna(null_value)
     elif dtype == "object" and null_value is None:
         exp[exp.isna()] = None
+    elif using_infer_string and dtype == "category":
+        exp = exp.fillna(False).astype(bool)
     tm.assert_series_equal(result, exp)
 
     result = values.str.startswith(pat, na=na)
@@ -342,20 +365,31 @@ def test_startswith(pat, dtype, null_value, na):
 
 
 @pytest.mark.parametrize("na", [None, True, False])
-def test_startswith_nullable_string_dtype(nullable_string_dtype, na):
+def test_startswith_string_dtype(any_string_dtype, na):
     values = Series(
         ["om", None, "foo_nom", "nom", "bar_foo", None, "foo", "regex", "rege."],
-        dtype=nullable_string_dtype,
+        dtype=any_string_dtype,
     )
     result = values.str.startswith("foo", na=na)
+
+    expected_dtype = (
+        (object if na is None else bool)
+        if is_object_or_nan_string_dtype(any_string_dtype)
+        else "boolean"
+    )
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected_dtype = bool
+        if na is None:
+            na = False
     exp = Series(
-        [False, na, True, False, False, na, True, False, False], dtype="boolean"
+        [False, na, True, False, False, na, True, False, False], dtype=expected_dtype
     )
     tm.assert_series_equal(result, exp)
 
     result = values.str.startswith("rege.", na=na)
     exp = Series(
-        [False, na, False, False, False, na, False, False, True], dtype="boolean"
+        [False, na, False, False, False, na, False, False, True], dtype=expected_dtype
     )
     tm.assert_series_equal(result, exp)
 
@@ -365,11 +399,12 @@ def test_startswith_nullable_string_dtype(nullable_string_dtype, na):
 # --------------------------------------------------------------------------------------
 
 
+@pytest.mark.filterwarnings("ignore:Downcasting object dtype arrays:FutureWarning")
 @pytest.mark.parametrize("pat", ["foo", ("foo", "baz")])
 @pytest.mark.parametrize("dtype", ["object", "category"])
 @pytest.mark.parametrize("null_value", [None, np.nan, pd.NA])
 @pytest.mark.parametrize("na", [True, False])
-def test_endswith(pat, dtype, null_value, na):
+def test_endswith(pat, dtype, null_value, na, using_infer_string):
     # add category dtype parametrizations for GH-36241
     values = Series(
         ["om", null_value, "foo_nom", "nom", "bar_foo", null_value, "foo"],
@@ -383,6 +418,8 @@ def test_endswith(pat, dtype, null_value, na):
         exp = exp.fillna(null_value)
     elif dtype == "object" and null_value is None:
         exp[exp.isna()] = None
+    elif using_infer_string and dtype == "category":
+        exp = exp.fillna(False).astype(bool)
     tm.assert_series_equal(result, exp)
 
     result = values.str.endswith(pat, na=na)
@@ -400,20 +437,30 @@ def test_endswith(pat, dtype, null_value, na):
 
 
 @pytest.mark.parametrize("na", [None, True, False])
-def test_endswith_nullable_string_dtype(nullable_string_dtype, na):
+def test_endswith_string_dtype(any_string_dtype, na):
     values = Series(
         ["om", None, "foo_nom", "nom", "bar_foo", None, "foo", "regex", "rege."],
-        dtype=nullable_string_dtype,
+        dtype=any_string_dtype,
     )
     result = values.str.endswith("foo", na=na)
+    expected_dtype = (
+        (object if na is None else bool)
+        if is_object_or_nan_string_dtype(any_string_dtype)
+        else "boolean"
+    )
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected_dtype = bool
+        if na is None:
+            na = False
     exp = Series(
-        [False, na, False, False, True, na, True, False, False], dtype="boolean"
+        [False, na, False, False, True, na, True, False, False], dtype=expected_dtype
     )
     tm.assert_series_equal(result, exp)
 
     result = values.str.endswith("rege.", na=na)
     exp = Series(
-        [False, na, False, False, False, na, False, False, True], dtype="boolean"
+        [False, na, False, False, False, na, False, False, True], dtype=expected_dtype
     )
     tm.assert_series_equal(result, exp)
 
@@ -692,36 +739,41 @@ def test_replace_regex_single_character(regex, any_string_dtype):
 
 
 def test_match(any_string_dtype):
-    # New match behavior introduced in 0.13
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected_dtype = bool
+        na_value = False
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        na_value = np.nan
 
     values = Series(["fooBAD__barBAD", np.nan, "foo"], dtype=any_string_dtype)
     result = values.str.match(".*(BAD[_]+).*(BAD)")
-    expected = Series([True, np.nan, False], dtype=expected_dtype)
+    expected = Series([True, na_value, False], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
     values = Series(
         ["fooBAD__barBAD", "BAD_BADleroybrown", np.nan, "foo"], dtype=any_string_dtype
     )
     result = values.str.match(".*BAD[_]+.*BAD")
-    expected = Series([True, True, np.nan, False], dtype=expected_dtype)
+    expected = Series([True, True, na_value, False], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
     result = values.str.match("BAD[_]+.*BAD")
-    expected = Series([False, True, np.nan, False], dtype=expected_dtype)
+    expected = Series([False, True, na_value, False], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
     values = Series(
         ["fooBAD__barBAD", "^BAD_BADleroybrown", np.nan, "foo"], dtype=any_string_dtype
     )
     result = values.str.match("^BAD[_]+.*BAD")
-    expected = Series([False, False, np.nan, False], dtype=expected_dtype)
+    expected = Series([False, False, na_value, False], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
     result = values.str.match("\\^BAD[_]+.*BAD")
-    expected = Series([False, True, np.nan, False], dtype=expected_dtype)
+    expected = Series([False, True, na_value, False], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
 
@@ -757,10 +809,17 @@ def test_match_na_kwarg(any_string_dtype):
     tm.assert_series_equal(result, expected)
 
     result = s.str.match("a")
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
-    expected = Series([True, False, np.nan], dtype=expected_dtype)
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected_dtype = bool
+        na_value = False
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        na_value = np.nan
+
+    expected = Series([True, False, na_value], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
 
@@ -785,10 +844,14 @@ def test_fullmatch(any_string_dtype):
         ["fooBAD__barBAD", "BAD_BADleroybrown", np.nan, "foo"], dtype=any_string_dtype
     )
     result = ser.str.fullmatch(".*BAD[_]+.*BAD")
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
-    expected = Series([True, False, np.nan, False], dtype=expected_dtype)
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected = Series([True, False, False, False], dtype=bool)
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        expected = Series([True, False, np.nan, False], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
 
@@ -796,10 +859,14 @@ def test_fullmatch_dollar_literal(any_string_dtype):
     # GH 56652
     ser = Series(["foo", "foo$foo", np.nan, "foo$"], dtype=any_string_dtype)
     result = ser.str.fullmatch("foo\\$")
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
-    expected = Series([False, False, np.nan, True], dtype=expected_dtype)
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected = Series([False, False, False, True], dtype=bool)
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        expected = Series([False, False, np.nan, True], dtype=expected_dtype)
     tm.assert_series_equal(result, expected)
 
 
diff --git a/pandas/tests/strings/test_string_array.py b/pandas/tests/strings/test_string_array.py
index 517ddb164985c..cd3c512328139 100644
--- a/pandas/tests/strings/test_string_array.py
+++ b/pandas/tests/strings/test_string_array.py
@@ -38,7 +38,7 @@ def test_string_array(nullable_string_dtype, any_string_method):
             expected.values, skipna=True
         ):
             assert result.dtype == "boolean"
-            result = result.astype(object)
+            expected = expected.astype("boolean")
 
         elif expected.dtype == "bool":
             assert result.dtype == "boolean"
diff --git a/pandas/tests/strings/test_strings.py b/pandas/tests/strings/test_strings.py
index 40b6c69dc8025..7c396e65b6120 100644
--- a/pandas/tests/strings/test_strings.py
+++ b/pandas/tests/strings/test_strings.py
@@ -217,8 +217,21 @@ def test_ismethods(method, expected, any_string_dtype):
     tm.assert_series_equal(result, expected)
 
     # compare with standard library
-    expected = [getattr(item, method)() for item in ser]
-    assert list(result) == expected
+    expected_stdlib = [getattr(item, method)() for item in ser]
+    assert list(result) == expected_stdlib
+
+    # with missing value
+    ser.iloc[[1, 2, 3, 4]] = np.nan
+    result = getattr(ser.str, method)()
+    if ser.dtype == "object":
+        expected = expected.astype(object)
+        expected.iloc[[1, 2, 3, 4]] = np.nan
+    elif ser.dtype == "str":
+        # NaN propagates as False
+        expected.iloc[[1, 2, 3, 4]] = False
+    else:
+        # nullable dtypes propagate NaN
+        expected.iloc[[1, 2, 3, 4]] = np.nan
 
 
 @pytest.mark.parametrize(
@@ -248,6 +261,7 @@ def test_isnumeric_unicode(method, expected, any_string_dtype):
     assert list(result) == expected
 
 
+@pytest.mark.filterwarnings("ignore:Downcasting object dtype arrays:FutureWarning")
 @pytest.mark.parametrize(
     "method, expected",
     [
@@ -258,10 +272,14 @@ def test_isnumeric_unicode(method, expected, any_string_dtype):
 def test_isnumeric_unicode_missing(method, expected, any_string_dtype):
     values = ["A", np.nan, "¼", "★", np.nan, "３", "four"]  # noqa: RUF001
     ser = Series(values, dtype=any_string_dtype)
-    expected_dtype = (
-        "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
-    )
-    expected = Series(expected, dtype=expected_dtype)
+    if any_string_dtype == "str":
+        # NaN propagates as False
+        expected = Series(expected, dtype=object).fillna(False).astype(bool)
+    else:
+        expected_dtype = (
+            "object" if is_object_or_nan_string_dtype(any_string_dtype) else "boolean"
+        )
+        expected = Series(expected, dtype=expected_dtype)
     result = getattr(ser.str, method)()
     tm.assert_series_equal(result, expected)