4AWG or 6AWG would be way more than enough for a car PC. 4AWG is a safer bet if going for a car PC + an amplifier.
As for the grounding choice, the pros of connection directly to the battery terminal are virtually none. You should however attempt to ground all audio components to the same point in order to avoid a ground loop. This includes the head unit and/or amplifiers for the speakers. Ground runs should also be kept as short as possible. You can likely get away with grounding everything separately to the chassis of the car but make sure you have good contact to bare metal at your grounding points.