Very few games don't. I have something like 260 games, including just about every "mainstream" title ever made. None of them have a problem working with the latest Swap Magic disk.
Not really. I've been doing the swap for at least four years now, and it hasn't bothered me at all. It's just one extra step, and isn't obtrusive at all. In fact, after awhile you don't even really notice it. That's not to say that you suddenly forget that you have to do it, but that whatever small problem it may be at first stops being one pretty quickly.Then there's the hassle of always having to swap discs when playing. You will get tired of this - you really will.
So have they worked out the overheating issues with slim-line modchips yet? I know that there was a problem with it that would cause your drive to go kaput pretty quickly. We're talking like within a half an hour of installation here.If you do want a modchip, I recommend either the MATRIX INFINITY of the DMS4. Both chips play about everything you throw at them (Swapmagic's compatibility is great too - but not SUPREME). Matrix should be a little cheaper than DMS, and is a premium ps2 chip.
If they've gotten over that problem, then great. I may even look into a mod-chip myself eventually.
EDIT: It looks like they've got a diode-fix that solves that issue. Still, they want almost 50 bucks for the chip. Add 100 bucks more if you want them to install it for you.