Well, some news. My computer failed to start three times, before finally starting. The third time, there was a post error message: One long beep, three short.
According to this, it means memory failure.
Upon finding my manual the beep code actually means that there was no VGA detected. So, this is almost certainly, if not more than one thing, a graphics card issue now.
Thoughts
Should I get a GTS 560i or GTS 570
(or something else?)
No on-board graphics...
Also, the first time I turned on, all fans (CPU, PSU, GPU) all spun at 100% and nothing happened - no error, no post, nothing. Don't know if this means anything about the motherboard (which I think is fine)