Instead of the trunk, how about under the passenger seat? I've thought of doing the same thing, and have found that all of the Antec SFF cases fit perfectly under there.
Depending on what you want to do, you don't need top end stuff and can get away with smaller items (like a little R9200SE-T
) and stuff it all in a really small case. This placement cuts down on the length or wiring and puts the PC in a climate controlled environment.
For power, you're going to need to worry about clean power and amperage. An inverter is a great idea and I've carried one around for a while now. I've ran dremels, laptops, drills, and small TV's off 'em. Mine is a 600W Sun unit with two AC outlets. You could hard wire this to a switched set on the fuse box (one that goes on and off with the accesories so it turns off when you turn off the key). This one is a bit large, but you can get smaller ones that could be stashed under the consol or under the dash. A relay and the blue amp signal wire is another option if you want anything to turn on or off with your head unit.
As for wiring the PC to the speakers, I'd get an aux input device for your head unit and use the soundcard/onboard sound minijacks. Some headunits still have a minijack input on the front for things like minidisc or portable CD players. Others have an I/O box you can add on to the back wiring to make an Aux out. Many of these boxes use RCA plugs, so you'd just need an adapter to mate it to the sound system.
For input, you say you'll use a keyboard and mouse. I say you won't need them once you get the thing running from touch screen. Windows XP has a built-in keyboard under the accessibility tab (may need to add this in "add programs"). It's a bit clunky, but it beats draggin around a keyboard. And again, you may not need it once the touchscreen is up. Same with the mouse. Keep 'em with you just for emergencies.