thanks for the PSA, I too have had similar problems with power cycling and needing to power things on in the right order. I think in my case it had to do with the TV not being able to read the EDID info from the video source correctly unless i turned the TV on first, and then my set top box.
The Xbox360 component cables can do 1080p (though lots of people think they cant). However,if it is an older projection TV I bet that the TV is the reason you are having problems with the picture quality. Good move switching TVs.