Update: Since the hard drive failed both Seagate's SeaTools "short test" and HP's own diagnostic tools, I have contacted HP for a replacement unit, as it's still under warranty. I ought to have this within the next few days.
However, I am still not fully convinced that the problem is solely with the hard drive. I would like to reiterate comments that I've made previously about what prompts the crashing. In every case to this point, one of the following is being done: playing a game (Civ4 and Galactrix), watching YouTube, or running iTunes visualizer and browsing with Google Chrome.
After the last post by mgorman, I started up the PC with the intent of running chkdsk a couple more times. However, it started up completely with no problems and has been running fine for the last two days, so I didn't want to rock the boat. This morning, though, I was watching a YouTube video -- more or less as a test. At exactly 9 minutes 15 seconds, the computer stuttered to a stop and crashed.
I haven't mentioned it before in this thread, but I've noticed this 9 minute interval with previous crashes. That is to say, after about 9 minutes of one of the causes listed above (game, YouTube, iTunes+Chrome), the computer crashes. My concern is that this is not the hard drive, but some other component, such as the video card.
After the latest crash, I tried restarting the computer numerous times; each time it crashed before getting to the desktop. I let it sit for ~40 minutes . . . and it started up fine.