Overheating will definitely cause your computer to behave in strange ways, especially if it is in an enclosed space or under a desk with a bunch of stuff on it.
I would look into getting a new power supply, the fan should be spinning and the power supply should NOT be hot to the touch. If you run it too long, you will start to smell that distinct smell of burning electronics....
And good troubleshooting, many people completely neglect the power supply as a source of the problem!!!
Remember, playing a game is more resource intensive than browsing the net... your hard drive is blazing, your processor is working over time, your memory is working frantically... all these things can cause the temperature in your case to rise... thus causing the problems you just stated.
You can get a cheap power supply in E-Bay, look around. Remember to get something with adequate power, a 300 or 350 watt PSU should do just fine.