I can't comment on the saving issue, as all my xbox gaming is still done on my xbox (why eat at the 360s laser if you don't have to) but I'd assume you could at least save the games to the hard drive if not the memory card. In fact i think that may be the only way, as i think you'd need the hard drvie to download the updates in order to play the newly supported xbox games.
As for the horror stories about overheating. I've had that happen once, and that was only because i left to get something to eat one night and didn't get back home till morning. Other than that I've never had any issues along the lines of the red circle horror stories, though I've had friends who swear that they had two 360s die on them in the span of a month, but as was mentioned, thats usually due to negligence on the part of the owner.
The must have games really all depend on what you want in a game yourself. As great a game as GoW was, once i finished it, and finished it again in co-op mode... I was finished with it. As wonderful as Oblivion is, its ever so much better on the pc version (particularly if you can output that to your HDTV with a dvi-hdmi converter) The games I've had the most fun with for 360 were:
Dead Rising (even if it gets repetitive, its still good zombie killing fun)
Lost Planet (one of the most unique and well designed FPS game I've ever played)
M:UA & Crackdown (random violence is always fun)
Guitar Hero II (its just better on the box.. but.. GH: Rock the 80s will make me get it for the ps2)
Chrome Hounds (can be completely insane online)
and the inexpensive as hell Kameo, which is still one of the more addicting platform adventure games out there
Note that all those games (save Dead Rising) are only made better by their online modes.
As for what model to get, that's really up to what you have, and what you want from it. If your tv has a hdmi input, its worth it to go and get the elite system, both for the hdmi support (particularly useful if you're gonna go the rout of getting the hd dvd player, as it needs the hdmi cable or a vga adapter in order to upscale your regular dvds) as well as the 120 gig hard drive (which if you're gonna use it for a lot of demos and videos, those can add up fast.
All in all the 360s a great system, and if you take proper care of it, it'll last you a long time. I could do without the Gold Subscription requirement to play games online, but its not TOO bad a price considering the crap that you get for free from Sony.
