UPDATE: While I think I might still have a "dead" CMOS battery, the main cause of the problem behaviors turned out to be CPU temps. One hex nut in the backing plate of the CPU cooler had come out, causing one of the 4 posts of the CPU cooler to not screw down, resulting in extremely high CPU temps (70 degrees c at idle). Pulling out the mobo and fixing this problem has brought CPU temps instantly down to where they should've been and I now believe the machine will be stable once more (still gonna try a fresh CMOS battery though).
UPDATE 2: Ordered a brand new battery, but CMOS settings are still lost on removing power even though everything else about the machine is now stable. Very odd.
My 8-9 year old Windows desktop has been acting really crazy the last couple of days, random shutdowns, refusals to boot/POST (and no beep codes), and most consistently, loss of CMOS settings on every cold boot (a boot after having removed all power).
So this sounds like a very straight-forward dead CMOS battery. But I tested with a voltmeter and the battery reads just over 2.9 volts, which seems like it should be plenty. I swapped it with another battery (and the only other one I have laying around) and it's also about 2.9 volts, no change in behavior.
Settings are not lost if I do a warm restart, or even a full shutdown without removing AC. I could order a replacement battery but this just seems so unlikely to me, but I can't find anything specific about CMOS voltages other than about clearly dead batteries.
Is 2.9 volts really low enough to cause loss of CMOS settings?