Wouldn't a nationwide bank run result in hyperinflation?
A bank run is an increase in demand for currency. Which would create hyperdeflation, as the value of currency rose to meet the clearing price at the current supply.
I don't follow you. If you will, please break it down.
All of those accounts are "insured" for up to $250,000. Where will this money come from? They will have to issue more money or default on their promises.
The Federal Reserve bank holds large quantities of currency. The Fed also owns securities and makes loans from which it profits. There is an awful lot of money available to fulfill demand without having to print more currency.
The Fed uses open market operations to do this. If the demand for currency increases (bank run) then the Fed offers currency to banks in exchange for securities (treasury notes, bonds, etc.)
The assumption that the Federal Reserve is also the Treasury is the common error. They are two distinct entities which have different ways to correct the money supply.