If this were a PC, I would say to open the case and clean out all the heat trapping dust because you describe classic heat related problems. But notebooks are not made to make normal, routine maintenance a user-friendly task, so you are stuck just making sure all the openings, cavities, and bays are clean, and that all fans are spinning. After cleaning, try blasting a desk fan across it. Also consider getting a Notebook Cooling Pad w/ext. power supply
I would also consider taking it to a shop for a professional cleaning.
You can test RAM using one of the following programs. Both require you to create and boot to a bootable floppy disk or CD to run the diagnostics. Using the floppy method is generally easier (and another reason to include floppy drives in new builds). However, the CD method is just as effective at detecting RAM problems. Allow the diagnostics to run for several passes or even overnight. You should have no
Windows Memory Diagnostic - see the easy to follow instructions under Quick Start Information.
MemTest86+ (for more advanced users) - an excellent how-to guide is available here.
Alternatively, you could install a single RAM module and try running with that to see if it fails. Repeat process with remaining modules, hopefully identifying the bad stick through a process of elimination.