Is the money that the questioner took from the bank—after an employee claimed ownership of it, even though it wasn't hers—considered unlawful money, and must it be returned to the bank, knowing that he has already given it away in charity?

If the amount is small, such that people of average means would not bother to seek it, then it is permissible to take possession of it without publicizing it or searching for its owner, and it is not obligatory to return it to the bank. The Prophet, peace and blessings be upon him, permitted taking and benefiting from small amounts, and this is the view of the majority of scholars. It is better to give it away as charity if possible, and he is not obliged to pay its equivalent if its owner is found.