Help me diagnose this system, really weird

Long story short, my wife and I purchased a computer for her daughter from a friend of mine because he needed to get rid of it and made us a great deal on it. System is a Biostar motherboard (not sure the model but it's got an SiS chipset and onboard video, Savage4 I believe) and it has a 1.3 GHz Thunderbird core using a heatsink that I THINK came off of an 1800+. Anyways, the computer is flakey as heck. It constantly gets blue screens, crashes, etc. I'm probably going to be rebuilding it soon (like maybe this weekend) but thought I'd run it past everyone first and get some more opinions on it. I work tech support for a living but this one still has me baffled. Here is a list of some information I've been able to gather on it.

• Get BSODs, sometimes when booting. The error tends to change a bit but at least some of them have suggested that it might be a problem with the hard drive or the file system (sorry, I don't have the exact error numbers as I didn't write them down and I stopped messing with it for awhile). If the system got a BSOD, it generally would not restart without getting another BSOD during boot. Sometimes powering it all the way down would let it boot, but generally not. If I turned the computer off for awhile, it would work okay for a bit.
• Per the above, I found that with the side on, the error seemed to occur more quickly which DOES suggest heat related.
• The CPU in the BIOS reads insanely hot, like 130C I believe. Obviously at those temps there's no way the thing should even be functional so I'm assuming that it's just a BIOS bug. The fan works and the heatsink does get QUITE warm, so it's obviously pulling heat off of it.
• I've flashed the BIOS with no change.
• I have tried adding a videocard in place of the onboard video since it would seem to lock up when, for instance, displaying a winamp visualization (the nullsoft ones that come with it FWIW) or watching a DVD. Still didn't help.
• I've removed the wireless PCI card, no go.
• I tried doing a repair of Windows and couldn't even get it to boot to the XP CD without getting an error. I swapped some memory and that seemed to resolve the problem for a short time. Memory wash Mushkin I believe and when I tried to RMA it, they're wanting me to go into the BIOS and crank the voltage and such. Note that the memory is 3200+ but is running at 2100+ speeds so is running way under stock and I don't see why I should have to push the memory above rated voltage to get it to work at all. So I bought new (and more) memory. That didn't really fix the issue either.
• At one point I realized that I kept getting a BSOD (DRIVER_IRQL_NOT_LESS_OR_EQUAL I'm thinking) as it loaded into Windows and played the startup sound. Disabling audio seemed to help there and I installed an old SB Live card I had (though I can't swear the card is still good since it's been sitting in the shelf above my closet for awhile and in a box for awhile before that. It's old, though, bought it when they first came out back around 95 I want to say.) That still didn't stabilize the system.
• I've run every diag I could find on the HDD and they pass.

I'm pretty much down to two possibilities and that is that either the motherboard or the CPU are having some sort of heat related issue. Could possible be the PSU as well I suppose (it's a cheapie generic that probably came with the case). I don't THINK I've tried replacing that though I might have.

So does anyone have any more ideas before I throw my old NF7-S and Mobile Barton 2500 in there? I'll probably have to do it by tomorrow evening at the latest.

Well, everything I was going to suggest was nulled when you said you tried it already. I'm down to the same conclusion you are: it's either the CPU/Motherboard, or PSU.

I really don't understand the temperatures part though. That has me baffled... maybe the BIOS is displaying the farenheit as celcius?

But yea, I think an upgrade to a Barton 2500+ would be a good idea anyway.