Am I obligated to ask my brother about the source of the Mushaf upon which "Waqf for the sake of Allah the Almighty" is written, and is it permissible for me to place it in any mosque to be absolved of sin, and if it becomes clear that he took it from a mosque, am I obligated to return it to the same mosque?

Donating a Mushaf (definition) as a waqf (endowment) is permissible, and it is not a condition that it must be endowed specifically for mosques. Rather, it is permissible to endow it for reading in any place. There is no obligation to ask the donor about the source of a Mushaf on which "Waqf for the sake of Allah" is written, as the default assumption regarding the actions of Muslims is validity. As for a Mushaf endowed for a specific mosque, it must be returned to it, and placing it elsewhere is not sufficient. However, if the Mushaf is endowed for mosques in general, then placing it in any mosque is sufficient.