The specs of the pc seem like it would do most things you would ask of a car pc, no probs there.
The thing used to turn off the pc when you turn off the car is called a shutdown controller (ITPS for example), search for "shutdown controller".
as for what you use to power the pc and screen...does the screen take mains power? or 12 volts? an inverter may be necessary because of the screen but the power solution of choice for a car pc is a dc-dc power supply (opus, arise etc.) these are more efficient and dont interfere with audio signals like an inverter can. Some models (opus) have a shutdown controller integrated negating the need for one.