Found the culprit after all. After tons of unplugging and swapping and testing. Somehow one of the 4 RAM sticks managed to break out of nowhere and my PC couldn't start because of it.
No idea how are the 6 beeps & the GPU stopping to work related to that problem (since ram errors usually give...
My DVI cable is working just fine (tried another one and it didn't fix the problem) and I switched my GPU from one PCI-E slot to another. Could they both be faulty?
I left my PC on for ~2 hours to download a game while I was away for some work. When I came back, the computer was still working but the monitor was not showing anything (black).
1. I tried to restart it because I had the same thing happening a few more times before and restarting usually...