The saga of my sad PC continues. Tonight, I replaced my old power supply with one Kleiner-One-Niner gave me (what a guy). I unplugged all the cables on my old one, unscrewed it, and yanked it out. I fastened the new one to the case, plugged all the cable back in, and hit the power button. Crackling sounds and smoke began to pour out of the case. I yanked the plug from the back and inspected the damage.
It turns out, I plugged a cable in the wrong way. It was a small, four-pin power cable that ran to my front-side sound panel. Normally, these plugs have guide wings that force the inattentive user to plug in the cable one way only. Sadly, this card doesn't have any and I was the inattentive user. When I plugged the cable in, I was off by one pin... and that one pin apparently didn't like being left out.
The worst part about all this is the power supply doesn't fix the problem I was having. The computer still randomly reboots and sometimes won't even post when being powered on. More investigation is necessary...