If the Federal government shut down on the east coast because of snow?

Yes, but Not from the East Coast.

No. Only those areas where the snow fall is so severe that the mail cannot be delivered will be shut down. Other places will have regular service.

Use your head. If you don't get mail it is only because the mailman is unable to deliver it. Just because you can't get your mail doesn't mean that no one else can. You wouldn't have to worry about this if you get a benefit and had direct deposit or a direct express card. So far in Michigan I've gotten my mail every day except on MLK day because it was a national holiday.

Of course not.

The government isn't the crazed monolith, stolen from a cheap scifi novel, that Republicans would like it to be.

The USPS is no longer a government controlled entity, they just need to pay fees to the government. And good luck getting your mail.

Well, nobody will be getting mail from the East coast.

A distinct possibility.

if a tree falls in the forest...