Searching google will yield a cheap ($50 or so) power supply for your laptop that will work nicely, straight from the 12-14V your car has. Using an inverter makes noise -- trust me
I used a laptop for my first carputer, and it worked really well -- Dell docking station allowed me to take it out easily when I needed to. VGA touchscreen should work fine. If you have a docking station port, chances are good that one of the pins is your "power" pin, that when grounded will act as your power button. Current-limited output for a power LED should also be around. You should be able to cleanly hardwire the 12V PS for your laptop through a switch. I installed the power switch, momentary power-on push button, and little amber power-on LED in my ash tray (to match the factory BMW lighting). This allows me to control soft-power to the laptop as well as hard power. Depending on your car's electrical system, leaving your laptop powersupply on (to charge the laptop battery or whatever) may or may not be a good idea. My bimmer has 2 batteries hooked up in parallel so I wasn't too worried, but a $2 switch is worth it regardless.
Your idea is very good -- you will run into the strangest of problems doing anything with a carputer, and this will let you deal with them before installing a bunch of strange hardware, then pulling it out to change software settings, then installing it again, ad absurdum.
Of course, running an old laptop will have certain problems, depending on how old it is. If you have a decent amount of RAM you should be ok for most things, but I wouldn't bother with anything less than 200Mhz, unless you're planning to run Lunix or Free/Net/OpenBSD and using some custom SVGAlib-based program for a GUI.
Good luck!
And if cops give you crap about anything, follow the excellent example set in Reservoir Dogs: pour on gasoline after cutting off the ear to the tune of "Stuck in the middle with you". (Ok, I just got a $140 ticket for having expired tags, I'm not exactly fond of law enforcement.)