I can't give you a complete answer, but I can provide some insight. My ancient desktop that ran WinXP (which was a piece of stuff) had the same issue sometimes. However, it had less than 1GB of RAM, a GPU that had less than 500MB, and stuffty cooling. The fact that it's not running explorer.exe could mean you have a virus; I'm just not knowledgeable to the point that I can tell you exactly what's wrong with it. The fact that you ran the start-up discs points away from the direction that your OS is corrupt.. Once again, maybe a virus.
Did you have antivirus software before this happened? You said that it was susceptible to viruses, but if it's just on the account that you're running XP (doesn't necessarily mean it's a virus in that case), and you did have antivirus software, how updated was it (if you had any?)
Maybe try reformatting with the install disc you were given?
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Oh joy.
Edit: Does anyone know how to run the actual windows explorer program from task manager/cmd because I can't find stuff about it online and no computers in my house are currently running XP.