I know this guy!
I talked to him at the Dell Customer Support Center last night. He reported he had finally diagnosed the problems with my PC, and had narrowed them down to "either the hard drive, the memory, or the mother board". I had, of course, already eliminated the fans and power supply as a potential cause of the problem. He will be shipping me the parts so I can continue troubleshooting on my own.
No, I am not making this up.