The suggestion from u/Dinosaurysus about the CMOS battery needing replacement sounds valid. Especially if you have the computer for a few years, and the erratic behaviour might be a sign of a dead battery. It pays little to try to replace it before resorting to more drastic measures. Just pay attention to buy the correct battery.

But I find it really weird that if I leave Windows on too long it reverts, but if I restart immediately: I can do whatever I want — except boot Linux, apparently, that won't work no matter what I do.

The first question: That's because the battery has some charge to hold the changes for a few minutes but not for longer? For the second question: I see you tried with two distributions, the first isn't exactly a major one, and the second one has certain requirements that must be met in order to run and it's not a livecd unless you tried an earlier version.

Replacing the battery is easy, but if you have been attacked by a rootkit that's vastly more difficult to address. Somewhat irrelevant to these two, if you want to experiment with linux you can try with something more suitable for general use from an established source.