If you have a 2001 model, it's on it's last legs and I guaranty the picture isn't nearly as sharp as it was 5 years ago., so whether or not you'd be able to get 1080P shouldn't matter too much.
from most TV owners' experiences these days, you're probably lucky the TV's lasted this long! Not a statement against Sony, but TV's in general.
Holy crap, are you serious? I've been thinking about upgrading to a new tv just for the heck of it, but now I think I'll stick with me free ~20 year old tvs until they die. Which might be after I do.