Like my first reply, I highly suspect it's something to do with power saving features. Setting it to never turn off the harddrive and disabling the screen saver are probably things I would have tried. I'm afraid I don't know what to do after that. There are sometimes a lot of power saving features in a computer, and not all of them can be adjusted in ways that I know of.
If it's a software issue, then changing the OS or drivers may fix the problem. If it's a hardware issue, then it probably won't matter what you install on it.
I find it interesting that it didn't crash while you had a game open, but does while playing music. I'm going to guess that the game loads the CPU significantly, and possibly other resources, while playing music doesn't. A lot of game loops will load the CPU even when they're not doing anything. Maybe try loading the CPU in other ways and see what happens. If keeping the computer loaded prevents crashes, then it's very likely a power saving feature problem.
Check on the CPU usage on the Performance tab of Task Manager (Ctrl+Alt+Del). Also make a note of extra CPUs/Cores, or Hyperthreading (not as good as an extra core, but similar). Doing tests where they're all relatively unused, and some where they're all loaded.
If you need a cheap way to load a CPU, just write an infinite loop in some programming language. I'm going to arbitrarily suggest Ruby, as I've been using it at work and it's a free download with a one click installer. Create a .rb file, and type this:
while true
# Do nothing
end