The Coil should be fine with an 18AWG wire. It pulls very little power and could easily use smaller wire.
The power feed wire will need to be sized accordingly to the power of your PSU and the length of the wire you are running. You can always check by using a volt meter. Check how much voltage drop there is from the beginning of the wire to the end.
In practice if you are under 10' and around 100 watts or under then your 10AWG wire will likely be big enough. If you are unsure then use an 8AWG wire. Too big a wire is not an issue but too small a wire is.
You can check voltage drop with a volt meter and it is simple to do. You hook the ground of the meter to a common ground point. Make sure your meter can reach both ends of the wire without moving your ground location. You then place the positive of your meter to the beginning of the wire and then at the end. Your difference should be very minimal if anything. Try it again with the computer powered up and the voltage drop should be the same otherwise the wire is too small. You should run this test both with the computer on and off with the vehicle in the same position. Running or not. The actual drop will be the voltage at the end of the wire compared to the beginning of the wire. The end of the wire should always be less than the front if not the same.