I have a Toshiba 40" HDTV, with PS2 component cables, I'm trying to get 1080i to work, but when I select it in the options it changes to 1080i, but when I press OK after the change, it reverts back to 480i.
It only runs in 1080i mode when your racing, and maybe the replays, but for everything else it stays in 480i.
Do a search for "HDTV" or "1080" and you'll find more information.
I don't quite understand how you got to your last assumption, but anyway...
When you select 1080i-mode in the menu, the game switches to 1080i, then asks you if the picture is okay. No matter whether you agree or you don't, it then changes back to 480i (if you want to play in 1080i, you obviously need to agree though). You leave the options menu and go race. As soon as the game has loaded the track (just before starting the race), it changes to the mode you've set, like 1080i in this case.
But what does that have to do with two player mode?