Somewhat, but to make that argument you'd have to show what percentage they're buying and how prices have moved. I don't think there's evidence of a large effect currently.

That wouldn't be conclusive, as one could readily make the argument that prices would have fallen without the Fed support.

The logic here is armchair economics 101: supply and demand. The Fed is propping up demand.

Econ 101 distinguishes between buyers with market power and buyers without.

You also need to distinguish between the expectations channel and the purchase channel. The Fed announcing it's going to pump money into the economy decreases expectations of a severe ongoing recession, which reduces the probability of default, which raises bond prices.

Yep! Some of the price and liquidity impact is the buying and some is just the announcement of buying. Levine had a nice column on it several days ago.