No, you can't. Car operates on 5V (~4.75) DC. Game GI is AC. It will likely destroy the car.

No, you need to find a 5VDC source to tap into. I would suggest a PC power supply plugged into the service outlet. When you start loading stuff on the 5V rail, things can start going wonky.

Yes. But use flux to make it easier and faster.

There must be a switch for the door, and maybe a second one for pressing the whole car. They are probably going to be very small and easily damaged. You could bypass that switch with a jumper.

However, I would be sorely tempted to add a switch to the playfield and hook it up in parallel with the car's switch(es) so that it turns on from a shot on the playfield.

We have some products for Matrix in the works that would help this, if you have can wait.

First is an AC to DC Magic Box converter.
GI Bulb with Matrix lead, box, output DC.

The second re Matrix Micro lights, like you see in Models, also plug and play, bright lights at first.
But they can be dimmed.
These should allow an always on light creation from GI, and additional lights, feature lit from Inserts.
As well as Matrix 12V Flasher connections too!

All are in Proto-type, samples flying back and forth, getting production and release, as soon as I can.

Art

You could try a DK pinball power tap, safe way to add a 5/12 volt power source for that and future mods too. Available from pinball life

1/2 wave rectify a GI lamp connection and use 3vDC to power it. That may be enough juice for it.

All you would need is 2 pieces of wire going to either side of a GI bulb socket and a rectifier diode for that. If you need instructions, lemme know and I'll draw up a diagram for you. This is how LED bulbs work.
http://www.allaboutcircuits.com/textbook/experiments/chpt-5/half-wave-rectifier/