This sounds like a hardware problem, I'm sorry to say.
First -
heat. When you have a fresh boot from cold, does it mostly run ok? If you do a reboot after a problem occurs, does another problem happen very very quickly? Heat problems could be related to the CPU or the chipset.
Second - You mentioned swapping RAM. Did you just swap the banks, or did you try NEW RAM? Try running with just one stick at a time. Better fans/heatsinks can resolve that.
Third - Disconnect as many extra devices as possible. Remove DVD drive, and any secondary PCI cards etc. If there is a fault in something, it can affect everything else by sending strange signals out, etc.
Fourth - Try a better PSU if your current "standard" one is very low in wattage output. 300w sometimes doesn't cut it these days, and different devices (on-board or otherwise) might be drawing too much power and there could be resource fighting!
The graphics card issues is very telling - black at boot time means that no software is causing the issues
