There is a small discussion about this in the 'Concern over laptop mount with airbag' topic, but I'll reiterate a few points. I've done this exact setup in both my personal vehicle and in the NSSL mobile mesonets that were used for the VORTEX 2 project (that I built). The problems that you mention are easily overcome depending on how you wire the computer. If you run the computer using an AC power supply, you can simply plug it into an inverter which you can turn off and on. If you opt for the DC power supply, which is much more energy efficient and slightly smoother running, you can wire the computer to your car's electrical system. Rather than attempting to wire the computer directly to the car's battery, I would suggest you run your own wires from the battery to a fuse box inside the vehicle, from which you can power almost anything you want. You could put a switch inline with this circuit so that you can turn it on and off at your will. Most of these computers can take a few fluctuations in power as you crank your car to start it, as long as it doesn't take do long or isn't a switch. If it's a problem, install a backup battery and use a UPS switch to move back and forth between the power sources. You could run the computer equipment briefly with the car off without too much concern of draining your battery, but I wouldn't let it go for too long. You could always hook it all up and do a few tests to figure out how much time you actually have (backup battery would eliminate this concern).
From personal experience, I might caution you towards using a touch screen. It sounds like a great idea, but was practically difficult to use while chasing. Plus, without paying an enormous amount, the screen will likely be fairly small. You could just run the computer to a regular monitor (14", 15', 17" ,etc) and use a wireless keyboard with a trackball. Just my suggestion from what I have done.