Does the FDIC's insurance of your money in your bank account cause inflation of the bank were to go under?
No, for three reasons. First, FDIC bailouts are funded via the guarantee fund, which is a pre-funded insurance fund that has never required an injection of cash. Second, if such an injection were to occur, it would happen through an appropriation from Congress, which would be implemented via a debt issuance from the Treasury— the Fed has zero authority to bail out the FDIC guarantee fund. Third, bank failures actually tend to involve a net contraction of the money supply, as distressed banks are less able to engage in new credit extension.