ok, your problem is hardware. That's it, no doubt, nothing else. Nothing in software can cause a computer that is completely shut down to start up. Why? Because there is no software. You can't run any problems with no power to the RAM.
What you've probably got is either a bad power supply or a fault in the motherboard, or a fault in the front-panel wiring. You have an ATX power supply (I'm certain), which means you have a soft power supply. IE, in old computer, you hit the power button and physical power was interrupted to the system. The circuit for main power was opened so nothing can flow.
Not so with ATX. With atx an electronic signal is sent to the power supply (I believe) which then stops all but some trickle power into the motherboard. When you hit the power button, another signal is sent to the power supply to reactivate main power. IE...rather than being a physical circuit, it's hardwired logic.
Hence why it might be the power supply. Anything else involved in that loop could also be at fault, but I'm rooting strongly for your power supply being bad. Otherwise, it could be the front panel connectors (ie, they short and thus 'think' you pushed power) or your motherboard is flaking.
Yay me. Try swapping power supplies with Mole's computer and see if the issue keeps happening.