Being an IT professional I can tell you, the absolute WORST thing about computers is trying to figure out a problem at your house without the proper equipment. For your list, the CPU is almost certainly not it, that should not be close to #1 culprit. I think I've seen maybe.. 2 actual bad CPUs, and they wouldn't boot.
The memory tester boot CD is a good place to start, but if it were bad memory, I would tend to think blue screens would be more apparent. (or restarting, which is what a lot of computers do when they get a blue screen).
The heating does some like it could be the issue, and as for it not freezing during the copying, it could be because it wasn't taking much to copy files. If you have on-board VGA, things like displaying graphics (screen savers, Mame, boot screen?) could cause the CPU to heat up quicker. As was said, as long as you aren't running an ANCIENT computer, the a temp gauge will be in the bios. The should be a selection in there for HW monitoring. When it freezes, and you reboot, go to the bios and check the temp (if it's overheating, it should be high). How quickly do you restart it after it freezes? What kind of intervals are the freezes? Say, if it freezes, and you restart, does it take a while to freeze again, or is it more inclined to freeze up more quickly. If it freezes up quickly after the first freeze, we have a pretty solid bet that it's a heating issue.
I would run a hard drive test (I'm honestly not sure what software out there can be gotten to do this, I use Easy Recovery Pro) but I would doubt it if there was a problem there.. which is good, because I HATE reinstalling windows. Bleh.
How long has the PC been trucking? Is this a recent thing?