1 - The PC is way overkill for in the car. An inverter is the only way you're going to be able to power it. Be wary of heat issues with that setup, as well.
2 - The shutdown controller has nothing to do with hibernation. That's a setting within the OS to tell the motherboard what to do on a power button press. The shutdown controller only simulates the power button press on the turn of the ignition key.
3 - Yes.