Does the mint just give the money it makes to the banks or do they pay for it in some way?

Answer 1

The mint does not give money to banks. Banks keep peoples funds safe and use those funds as investments which in turn pay interest for the people keeping their money there. Banks keep the bulk of their funds in federal reserve banks. (They do not keep all the cash that people deposit etc.) If you are talking about replacing old bills, yes, the mint destroys old bills etc and replaces them with new ones via the federal banks.

Answer 2

The new money which is printed by the mint is made to replace the old worn out money which is picked up and (I believe) recycled.