The Series.str.contains method expects a regex pattern (by default), not a literal string. Therefore str.contains("^") matches the beginning of any string. Since every string has a beginning, everything matches. Instead use str.contains("\^") to match the literal ^ character.
To check every column, you could use for col in df to iterate through the column names, and then call str.contains on each column:
mask = np.column_stack([df[col].str.contains(r"\^", na=False) for col in df])
df.loc[mask.any(axis=1)]
Alternatively, you could pass regex=False to str.contains to make the test use the Python in operator; but (in general) using regex is faster.