A fairly positive diagnosis has been made as to what has been wrong with my computer. Here's a picture of some capacitors on the motherboard.

The bottom three, if you couldn't tell where I was going with this, are not supposed to look like that. The electrolyte inside them is all oozed out and crystallizing on the outside. (Batteries do this sometimes when they get old, if you've ever noticed similar guck on the inside of a little-used flashlight or something.) My dad starts off describing this with "This could cause data loss--" Well no shit! But fortunately, this fits all the symptoms: onset being at a random time that didn't seem to have anything to do with anything, getting progressively worse, the fact that for the past week or so I had been getting random (both in timing and in supposed responsible file) STOP errors all over the place, but nothing could find any disk or memory errors, and that when I was trying to reinstall Windows as part of a long saga I won't go into here, it kept erroring and not being able to copy files correctly. It also means that the processor and the memory (the latter being more valuable) are probably just fine and could be used in another motherboard (if it were possible to buy P3 mobo's anymore, which, well, good luck - but that's another problem).
What computer am I using now, you might ask? My dad had another almost identical (but not quite) motherboard which I swapped in, and then pulled over the full backup of C: on D: that I had done right before I started on the aforementioned omitted long saga (because I just knew I would need it) with its Automated System Recovery disk. (tell me, why does it have to format the drive and install Windows just to clone back a backup, when that Windows is only going to get written over anyway? wouldn't it be smarter to run that all from the CD?) There were a couple little casualties of this maneuver, though, the most annoying of which is that it somehow got to the Standard HAL instead of the ACPI HAL - and Microsoft's recommendation for this is "reinstall Windows" and it won't let you do it any other way, at least not from the GUI (in Win2K you can switch HALs from a list in Device Manager, but the KB says "don't do that, asshole; you WILL fuck up Windows"). The only other option is to use Recovery Console to re-extract the right HAL and the right kernels from the CD, which I may try, but it almost sounds easier and less hair-pulling to just really reinstall Windows, even though it means re-installing updates (I have a CD with SP2 on it, so that's not really as big of a pain as it sounds like, not as much downloading of the world as it was the first time).
Meh, anyway. In the meantime it runs. The real problem is that I can't keep this motherboard - it's one my dad bought to make himself a computer similar to mine (the toasted one, that is). If I keep it, I have to buy him a new one, or else I have to buy myself a new one. And then of course there's the problem that I can't just merrily install and activate Windows on a whole 'nother hard drive...