After doing my reinstall the problem didn't immediately manifest. It wasn't until the following day. The only thing I did was install all drivers (windows update and computer website for those that WU didn't privide), install Malwarebytes, MSE, CCleaner, OpenOffice, Guild Wars 2, and League of Legends. This setup lasted around a full day, then the problems returned after a Windows Update that wasn't available the first day and a restart. On rolling back to before those the problem stayed, and on loading my clean reinstall image the problem stayed. I still don't know if it's driver or hardware and I don't know what diagnostic tests would show. There still hasn't been any BSoD.