I think that was meant to be RGB cable? Aka scart, but not common in the US.
Composite I wouldn't recommend for anything, I find a great difference between that and component even for the PS2. If your TV has two compenent inputs, just go for both PS2 and Wii on component. If it has one, you can get a switch. Or go with one of them on S-Video (about scart quality). It's not really about the HD in the case of these two, to me component just handles colours better.
PS3 and Xbox360 best with HDMI. TV should have enough inputs I think. Mine is like 2 years old and has 2 HDMI.
38" I think is alright to connect everything to. I've read some people complain about PS2 looking bad on huge screens, but with my 32" TV I find it better than it was on the regular TV.
No reason to connect the Wii to the old TV since it has decent widescreen support.