I have an older PS2 Connected to a 40" Samsung HDTV that I have in my dining room. I play games in "GAME MODE" which is a lower resolution. It doesnt look that nice, but it might just be the PS2 is nowhere near full HD. I am just using the R L Video RCA style plugs.
A PS2 will look awful on a modern TV, especially the closer you are to it. And using composite to boot? lol Typically game mode means it doesn't do any processing, so there is no latency. I don't think it has anything to do with scaling down the resolution.
When I got my PS3, I used to just play San Andreas on it, which was a PS2 game. It didn't look great, but on a couch 10ft away it looked good enough.
Playing PC games on a 1366x768 TV will be fine, as long as you are maybe two feet or so way, as
BrightCandle said.