Something is definitely not right with your system. One of my two Athlon 64 systems is a 3700+ and at idle it runs at 26 deg C, and when under load at about 30-37 Deg C. So if your CPU is running at 60 there is something not right at all. Either the motherboard isn't reporting the correct temperature, or the CPU is getting much too hot. My one is also using the stock cooler that came with the retail CPU so nothing special. I would definitely recommend removing the heatsink, cleaning it up and applying a very thin new layer of thermal compound, as it doesn't sound like the CPU cooling is being that effective.
The other problems that could be causing such issues include the motherboard, ram or PSU being faulty. Instabilities you are experiencing are common with ram related issues. Have you tested the ram? And have you also tested that there are no bad blocks or read errors on the HD?
The first thing I always check when I have an unstable PC is the System logs (right click my computer, select manage, go to System tools/Event Viewer/System) and look for any red X entries or yellow exclamation mark entries and look up the errors thay are giving in google.

