1) Ideally, you don't want to use AC currect. The car provides DC. The computer uses DC. Rather than convert DC to AC and back to DC, you're better off with a DC power supply for your laptop. They aren't all that expensive ($50-75 or so) and far more efficient than using an inverter.
2) To pull the laptop out of hibernate, you typically press the power button. Some computers (desktops, mostly) have a BIOS setting that will allow boot when they receive power. You need to look in your BIOS to determine if yours has this feature, but I seriously doubt it does.
You've stumbled upon the most significant hurdle for using a laptop as a vehicle PC: Turning it on automatically when the car starts.
Do some searching around the forums and you'll see some pretty creative solutions.