Is the loan that the state repays to the bank with interest considered usury (riba), and is it permissible for me to take it?

If the loan is a good loan (qard hasan) without interest, then there is no harm in taking it. The fact that the lender obtained the money through a usurious loan does not harm the borrower, because the borrower owns the loaned money and is permitted to dispose of it, while the sin falls upon the lender.