So I gradually put everything back in the case. No problems all the way through. Then I stress tested everything simultaneously for over 8 hours. AIDA64 got a memory access violation error message at some point, but as that's a massive improvement from the sad state of affairs before, I'm not too worried about it. I guess I'll never really know but I think it was a combination of a loose pci-e slot and minor inaccuracies in the construction of the case. When I was testing the system outside the case the graphics card just leaned off to one side and worked perfectly, so I figured I'd position the motherboard so as to replicate that exact positioning as closely as possible.
When I put it in I noticed that no matter how I positioned the motherboard, the screw holding the graphics card would always apply a small amount of stress to it and, potentially, pull it an extremely minute distance out of the far side of the slot. The case is mostly a tool-less design and has clips that are supposed to hold the vid card in, but due to bad experiences in the past, I had opted to pull the clip out and use a screw instead right off the bat. Someone who reviewed the case on newegg said they had done just that because the clip was not as tight as it could be. I gave the clip a chance though and it appeared to put no stress on the card.
I didn't know a loose graphics card could cause such a wide variety of errors. Or that it was even close to that sensitive to position. We're talking on the order of less than a mm worth of lateral movement at the screw hole. I've never had to go to such care with installing an expansion card before...
All's well that ends well I guess...
