Thanks for the info Justin.
So at this point, this laptop is either hanging on a blinking cursor, or screen-scrambling after the BIOS screen and prior to Windows loading. I managed to get in on safety mode once, but the moment after I opened Chrome the screen scrambled.
I know this model and this laptop does have an issue with overheating, and I'm definitely not overclocking. It looks like the BIOS is two versions out of date, but trying to flash it with this rapid random BSOD seems like a good way to end up with a bricked motherboard. I don't think I'll be able to pull off any more than the 5 or so minidumps I managed to get off there a few weeks ago (which I'm guessing show the same thing in each). Is there nothing else to go on from the minidump analysis that might help me figure out how to resolve this?