Certainly seems like a hardware issue, but there's no easy answer to this one. It could be your power supply, motherboard, CPU, memory, even the video card. The power-button issue makes me lean towards power supply or motherboard though (assuming you've verified the power button works after a fresh boot).
If you have other parts on hand (even from another running system) you could swap components until you identify the culprit. If you find it's your power supply, make sure you replace it with a decent quality one, NOT one of those $25 units you find everywhere, or you'll have even more problems followed by a rapid failure in another year.