So, it turns out I treat my computers much like... oh, a parent that leaves their children in a car with the windows rolled up in the hot summer sun while they go into a strip joint, i.e. terrible.
Since the computer wouldn't even remain running long enough to bring up the Windows Recovery disk, I figured I'd actually check the hardware components again. (That will learn me to expect a certain type of failure to behave in a certain pattern!) The fan on the processor was so full of dust that it was struggling just to turn, and I thought I almost burned myself touching the metal on the processor. I'm pretty lucky I didn't lose the machine in its entirety to a fried processor due to my inhumane negligence.
I grabbed the vacuum cleaner and a bendy straw and sucked that fucker like a $2 whore. Things are running great now, and I think the server will be fine. I got the heat sink, the power supply and everything else I could get at with the straw.
Hopefully my server woes are over, for now, but at least I was able to identify and rectify the issue... this time. Comcast continues to suck, of course, so the site may be spotty even still until I figure out what the hell's wrong with them. But the server's going! Yay!