most of these questions can be answered by heading over the FAQ Emporium
you might be able to put the eeepc in the glovebox, and run the wires from there, or under the passenger seat.
if you are planning on keeping the stereo, run the pc through the stereo, the easiest way would be through the aux input, if your stereo has one.
if you want to get rid of the stereo, than you would run the laptop directly to the amp, but-- i have a 01 eclipse, and the factory amp was not adequate after removing the factory head unit(not sure how close the cars are). i would highly recomend getting a aftermarket amp to power all the speakers, this way, it give your one more point to adjust the system to sound better.
for the static issue, it might have something to do with the power adapter that you are using-- i'm don't have a laptop install, so you might need to look in the power supply, or laptop area.