The Federal Reserve Bank's current balance sheet (as of last Thursday, https://www.federalreserve.gov/releases/h41/current/h41.htm ) shows they hold $4.5 trillion in Treasury notes and bonds and $2.6 trillion in MBS (mortgage-backed securities) out of a total of $8.3 trillion. The first two numbers are face value not market value.

If market value is 10% less than face value for these securities on average, then the Fed would have unrealized losses of about $700 billion. But the Fed doesn't have the same insolvency or liquidity risks that ordinary commercial banks do.

The US dollar would go down in value if there were more dollar sellers than buyers. That would happen if those with dollars had something else to buy. The Us dollar index is up a bit today, so far. I'd imagine at the moment that a run on the dollar would be unlikely. But there could be some combination of circumstances . . .