diff --git a/pandas/core/algorithms.py b/pandas/core/algorithms.py
index 77cd73fdfe91b..e7ec7fde762f4 100644
--- a/pandas/core/algorithms.py
+++ b/pandas/core/algorithms.py
@@ -8,7 +8,9 @@
 from textwrap import dedent
 from typing import (
     TYPE_CHECKING,
+    Hashable,
     Literal,
+    Sequence,
     Union,
     cast,
     final,
@@ -27,6 +29,7 @@
     AnyArrayLike,
     ArrayLike,
     DtypeObj,
+    IndexLabel,
     Scalar,
     TakeIndexer,
     npt,
@@ -1338,10 +1341,12 @@ class SelectNFrame(SelectN):
     nordered : DataFrame
     """
 
-    def __init__(self, obj, n: int, keep: str, columns):
+    def __init__(self, obj: DataFrame, n: int, keep: str, columns: IndexLabel):
         super().__init__(obj, n, keep)
         if not is_list_like(columns) or isinstance(columns, tuple):
             columns = [columns]
+
+        columns = cast(Sequence[Hashable], columns)
         columns = list(columns)
         self.columns = columns
 
diff --git a/pandas/core/frame.py b/pandas/core/frame.py
index 5f4207d0985ef..36e8177477559 100644
--- a/pandas/core/frame.py
+++ b/pandas/core/frame.py
@@ -1374,7 +1374,7 @@ def dot(self, other: Series) -> Series:
     def dot(self, other: DataFrame | Index | ArrayLike) -> DataFrame:
         ...
 
-    def dot(self, other: AnyArrayLike | DataFrame | Series) -> DataFrame | Series:
+    def dot(self, other: AnyArrayLike | DataFrame) -> DataFrame | Series:
         """
         Compute the matrix multiplication between the DataFrame and other.
 
@@ -2155,7 +2155,6 @@ def maybe_reorder(
             to_remove = [arr_columns.get_loc(col) for col in arr_exclude]
             arrays = [v for i, v in enumerate(arrays) if i not in to_remove]
 
-            arr_columns = arr_columns.drop(arr_exclude)
             columns = columns.drop(exclude)
 
         manager = get_option("mode.data_manager")
@@ -4383,7 +4382,13 @@ def predicate(arr: ArrayLike) -> bool:
         mgr = self._mgr._get_data_subset(predicate)
         return type(self)(mgr).__finalize__(self)
 
-    def insert(self, loc, column, value, allow_duplicates: bool = False) -> None:
+    def insert(
+        self,
+        loc: int,
+        column: Hashable,
+        value: Scalar | AnyArrayLike,
+        allow_duplicates: bool = False,
+    ) -> None:
         """
         Insert column into DataFrame at specified location.
 
@@ -4396,8 +4401,8 @@ def insert(self, loc, column, value, allow_duplicates: bool = False) -> None:
             Insertion index. Must verify 0 <= loc <= len(columns).
         column : str, number, or hashable object
             Label of the inserted column.
-        value : int, Series, or array-like
-        allow_duplicates : bool, optional
+        value : Scalar, Series, or array-like
+        allow_duplicates : bool, optional default False
 
         See Also
         --------
@@ -6566,7 +6571,7 @@ def value_counts(
 
         return counts
 
-    def nlargest(self, n, columns, keep: str = "first") -> DataFrame:
+    def nlargest(self, n: int, columns: IndexLabel, keep: str = "first") -> DataFrame:
         """
         Return the first `n` rows ordered by `columns` in descending order.
 
@@ -6673,7 +6678,7 @@ def nlargest(self, n, columns, keep: str = "first") -> DataFrame:
         """
         return algorithms.SelectNFrame(self, n=n, keep=keep, columns=columns).nlargest()
 
-    def nsmallest(self, n, columns, keep: str = "first") -> DataFrame:
+    def nsmallest(self, n: int, columns: IndexLabel, keep: str = "first") -> DataFrame:
         """
         Return the first `n` rows ordered by `columns` in ascending order.