I'll get some questions addressed at least. Note I don't have an HDTV myself yet, but I've been looking into it and would like to get one in the near future for the X360/Wii and the PS3 later on down the road.
1. 720p is better for games simply because it's supported more. All X360 titles support this (though I'm hearing it'll upscale the games to 1080p now, but better to take no chances I think), and it seems like all PS3 titles will support it, with 1080p being a maybe. Also, the resolutions are 480i (SDTV. basically every NTSC TV made before HDTVs), 480p, 720p, 1080i, and 1080p. i stands for Interlaced, and p stands for Progressive. Interlaced basically means it does half the picture with every other line, while progressive means it's rendered all at once: I'd google up pictures or gifs showing the difference, and hit wikipedia to find a detailed explaination on the differences.
By the way, PS3 = Support up to 1080p natively, will support 720p, but downscale 720p games to 480p on a 1080i display, X360 = support 720p natively, will upscale to 1080i/p, and Wii = Support 480p, may have games with 480i (like Raving Rabbids I hear).
2. Downscaling/Upscaling basically means that a picture of a different resolution has it's resolution changed, whether by the internal TV hardware or the systems themselves, to match a specific resolution, basically whatever the TV's actual resolution is. Obviously, downscaling means the resolution is lowered, and upscaling is vice versa.
3. Component and HDMI are the ones to look for for HD gaming. For some stupid ass reason Sony, for all their pushing the HD era, threw composite cables with their system: However, it still uses the same output as the PS1/2, so get PS2 component cables and you should be fine. Or look for cheap HDMI cables online with Froogle, MySimon, or eBay. Only the PS3 has HDMI output, the X360 and Wii only have component, though the X360 may have a hardware revision with HDMI in the future, but so far that's just rumored.
4. I'm hearing that, despite what was said before, the PS3 will upscale the resolution of PS1 games, or just do something with the image so that they will look great on an HDTV, unlike on a PS2 with the same connections.
This is something you /will/ in all probability want to thoroughly research, and not take just one person's word on. HDTVs, when upscaling something like 480i, will have lag as the image is processed. It's a few milliseconds, and likely won't matter if you're just watching something or playing a game that doesn't depend on reflexes. But this lag is enough that it can royally fuck you over for anything that requires split second timing, like Guitar Hero or fighting games. I believe GH2 has a lag option, so that you'll be covered if playing on an HDTV, but obviously the vast majority of old fighters. Maybe it won't matter for you, but if you're that serious I would at least seriously look into it, and worse comes to worse keep around your current TV for playing the fighters on.
5. I hear that plasmas eat up a lot more energy, and will slowly burn out over time, and you're pretty much screwed when it comes to fixing them. They also only come in large models, while LCDs come in smaller sizes. You will still get huge LCDs though, and I believe the gap in size is closing between them. And if you mean CRTs with the big tube TVs... Yes, there are. But they only support 1080i and 480/540p from what I understand... And while this would be fine for the X360, the PS3 will just go 'wtf' if the game can't support it and downscale to 480p. So for the PS3, I'd forget about them.
I hope that helps, and check these forums too