1) Always check readmes, if you don't already. Make sure games and emus are in their default locations if they have one - sometimes this can cause issues, although you'd have to have very bad luck for it not to work on as much as it sounds like is the case.
1a) Unlikely from the crashes you describe, but check your batteries. Make sure they're freshly charged NiMH ones with a decent capacity (say, 2500mAh). If you are using Alkalines, then stop doing so and buy some rechargeables. The GP2x runs for very little time at all even on High Capacity Duracells, let alone on any others. That said battery-related crashes usually involve a white screen with no response to button input, multicoloured lines after a while followed by cessation of response, crashing when overclocking, or randomly turning off. Not black screens or being kicked back to the menu.
2) Return to menu indicates that the programs in question are crashing. Is there any way for you to test your RAM's integrity? If it's gone screwy that would potentially cause programs to crash, which would, of course, go back to the menu (or in those cases where the program explicitly tells the 2x not to relaunch the menu, a black screen). Nb that you should check this before the next point because IF your ram is buggered, then any firmware image loaded into it will get corrupted, so you could conceivably end up with a brick. I would imagine if this is the case you'll be able to ask for a replacement 2x from whoever you bought it from.
3) What firmware are you on? I understand you have an F200, so you should either be on 4.0 or 4.1 (assuming 4.1 exists for the F200). Even if you *are* I would recommend a re-flash - it's conceivable your firmware is just slightly corrupted. As I say, check the ram before a reflash, just in case.
4) If anything is working, what is? Are any working programs those with the mmuhack.o object? If there are, are those programs that are not working those with the mmuhack.o object? If there's a clear pattern, try deleting the mmuhack.o if those with it aren't running. NB that this will mean everything will run much slower than it should, and may indicate that there is a problem with your ram as speculated in point 2. If on the other hand the only programs that work are those with mmuhack.o present, you should try applying it manually in a script before running the requisite programs. I'm afraid I don't know how to do this.
6) Equally, are there any problems with video playback? Video playback uses the second CPU afaik, so if there are problems with that and with other programs that use the second CPU (Picodrive springs to mind, but there are others), that are not traceable to mmuhack.o (i.e. deleting it doesn't change status for that program), then you may have a faulty MMSP2 (the CPU) with a problem in the second core.
Can't think of much apart from those. If you *are* following readmes correctly, and you're using fully charged batteries, then almost any of the other possible issues are manufacturer faults, and mean you should ask for a replacement GP2x. The only one that isn't is the firmware-is-slightly-corrupt-but-not-too-badly, but as I say, find some way to check your memory first before you try reflashing, just in case. There *may* be tests to help in the CPU/LCD tweaker, but I don't know how well they work.