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.
They will default on their promises. That is why you either want to have your cash in your mattress, or in a very strong bank.
Or, most cynically, put your money in a very weak bank. If the banks start going down, you want yours to go down first when they're still paying on the 'insurance'.