As is often the case with computers, I have fixed mine while having no real idea what I did. My last resort apparently worked, but since that was "pull everything out and then plug the essentials back in, then work up" I'm not sure which bit fell into place in the process. My best guess is that something important was loose and got properly secured when I shoved it back into place. Either that or it really did matter which RAM set went in which slot, as they wound up in the reversed slots when all was said and done. (Except that never mattered before...)

My graphics card fan is still making funny noises for about 20 seconds after bootup, but it's been doing that for a while without causing problems, and I eliminated it as a cause by trying other cards. So who knows?