Chkdsk ran for 3 days (!) and then the computer rebooted spontaneously - no idea why, maybe there's a limitation on the Windows setup disk I used to run it. Anyway, I *was* able to rescue almost all of the data (I lost about 20GB of the 2TB I was using, but nothing irreplacable) and set up software RAID1 a few days ago - but my spurious freezing problem seems to have gotten WORSE (the high I/O load is probably triggeting it somehow). I have yet to be able to complete synchronizing of the drives, it keeps freezing after a couple of hours. I don't want to use the computer too much until that sync is done.
Today, I've tried both replacing the motherboard and running on the integrated GPU instead of a separate one, with no luck. I am currently trying one stick of RAM at a time (despite neither stick showing any errors in MemTest, even during the 30-something passes I ran when the problem started happening); the second (and last) stick is being tested right now, and I fully expect it to freeze again some time overnight (but I really, really hope it won't). (EDIT: it just froze as I was going to bed. Oh joy.)
I know it's not a software issue; I had these freezes happening even when I was copying data around with a Linux boot disk to salvage the data and prepare the partitions for Windows RAID. So, assuming this stick of RAM also fails, that only leaves the CPU as the possible faulty component. I really don't want to have to dig even further into my savings (I have 0 income currently, and already spent about 400€ on parts these past two weeks since it failed), so I will likely have to contact Intel and ask about getting it replaced under warranty, and I have no idea how long that might take. I might be able to turn off some things in the BIOS to get it working better until that can happen, but I can't make any promises.
If that *also* doesn't work, then I literally have no idea what to do. At that point everything except the case and a Blu-Ray drive would have been replaced or tested in isolation.
tl;dr: computer is still not working properly, no ETA on a fix.