Random stuff usually means hardware. The primary suspects: heat, as you mentioned; bad capacitors on the motherboard; power supply; and failing memory.
Heat seems less likely in this case because the problems commence as soon as you turn on the PC. However, it's easy to check. Do you know how to get into BIOS SETUP on your PC? If you can do this most modern motherboards have a page where you can look at the temperatures. Any CPU temp over 50 C with the system idling is a red flag.
If you don't know how to get into SETUP please check your manual or post back with the make and model number of your computer, we'll try to help you.
To check for bad capacitors go to http://www.badcaps.net
and look at the pictures. Take the side off your case an take a close look at your mobo. Use a flashlight and magnifier if necessary. The swelling can be pretty subtle.
Memory can be tested with a couple of programs. One of them is available from Microsoft here
. Be sure to read all the instructions on the page. Also please note: Do not use your malfunctioning machine to download and build the boot floppy disk or CD! Do it on a working computer!
The only way I know to test a power supply is by substitution. For that reason I suggest checking the other things first. Good luck.