The ultimate problem was the attempt at doing wireless HDMI to the projector -- which I thought was necessary, but turned out not to be.
One evening I noticed that bottom of my front door wasn't flush with the floor. There was an inch of weatherstripping that created clearance between the floor and the bottom of the door.
That was the magic realization that told me I had a route for an HDMI cable from the AVR to the projector.
I bought a 35-foot cable from Amazon, and rewired the system the way it was intended to be: HDMI from all of the sources, MONI 1 to the plasma, and MONI 2 to the projector. No other gimmicks in the chain.
Everything worked perfectly immediately.
It's possible that I could make the wireless HDMI kit work by using the Gefen HDMI Detective: If I put the Detective box in line ahead of the wireless transmitter, my guess is that it would provide a stable EDID for the AVR, and make it stop flailing about as the wireless kit changes state. I think this was the ultimate cause of my problems.