I can't speak to your HDTV questions, but I can make a recommendation for your Neo Geo, Saturn, and all older game systems: use an SD display.
It may seem counterintuitive but HD is not always the best solution, and it's certainly not for old, low-res video games. The vast majority of console games before the Dreamcast appeared ran in what might be called 240p, the form of 480i that looks progressive and has scanlines. Systems like Neo Geo and Saturn displayed their graphics at 320x240 or below, and were intended to be viewed with scanlines. The scanlines soften the jagged edges of these low-res images so that they actually look pretty good.
In comparison, an HDTV (and fixed-pixel displays in general) cannot display these low-res signals in their native format; it has to process and modify the image, which always means line-doubling and usually means additional effects which vary by HDTV model. This results in a 480p image devoid of scanlines. In their place are duplicated lines, so each pixel looks twice as tall. It's akin to viewing a game on a PC-based console emulator. Along with motion shearing and other possible artifacts, the end result is not pretty in my opinion, though it's definitely a judgment call.
In addition, the HDTV's signal upscan process runs the risk of introducing processing time, which will be noticed as input lag. Obviously, this is to be avoided at all costs, and varies by HDTV model.
I could not live with the compromises inherent in retro gaming on an HDTV, so I sought out a superb RGB-compatible Sony CRT (KV-29DS65) that I finagled out of Japan. Analog RGB was the best signal type possible before Component appeared, and is what arcade games use (or used to). With my RGB setup for old games and soon a solid HDTV for Xbox and later, I'll be all set.
For more info on retro game RGB stuff I recommend the GamesX forum (check Google). It's down at this specific moment, but should return whenever. Essential resource.
Edit: I should add, there is a possible middle-of-the-road solution. You could use the Micomsoft XRGB-2 (described in this thread's first post) to upscan your old consoles' 480i/240p RGB to 480p VGA. The XRGB-2s have a "fake scanline" feature that draws black lines over every other line. It's a fair effect, better in the later XRGB-2+ models than the old non-plus ones. It's not perfect, but could well be good enough to get your game on.