1. It will be good for music and GPS. I don't think it'd work well with movies (anime) unless you have an external video card. The 733MHz I have in my sig (leftmost one) was running music and GPS just fine.
2. You have a choice of either running mini-to-RCA to the headunit (your car's radio) or mini-to-RCA to amp (thus controlling the volume of your music via the carputer).
3. Wires are never welded! At most, you'd solder them, but that too doesn't necessarily apply to your situation. You just have to extend the wires from the Power Switch on the motherboard to whereever you feel comfortable.
4. Wiring to the battery = Power. No computer is going to work without it.
If you shop around and purchase the components wisely, you should be able to have everything for around $750. <-- That's just a rough estimate based on your requirements and other people's similar setups.