1. your best bet is to buy a DC-DC converter (look for it under "MP3Car Hardware" on the main mp3car.com web page. it'll be slightly more expensive than a DC-AC inverter, but its worth it. a power inverter that plugs into your cig lighter will work, but if it's for show then thats not what you want to do
2. depends on the screen. If its an RCA/NTSC model then you are gonna need an RCA out card (they cost about $30). from there its real easy: go to radio shack and buy a sheilded RCA cord and fling that sucker to the front of the car. If you want to use a laptop screen then you have to buy a $200 controller card from www.flat-panel.com and a VGA exstension cord (im not sure where to get these or how much they are).
3. Depends on your head unit:
If you have an AUX or CD-IN jack, use that. you get the best sound quality with this method.
There is a way to hook into your amp, but i cant recall the wiring and you indicated that you did not want to do this (for technical reasons).
You can also get an FM modulator from www.partsexpress.com which loses some sound quality, but usually only the true audiophiles notice. it runs $50.
the least desirable way, but by far the cheapest and easiest, is to use a cassete tape adapter. you CAN do this, but I dont suggest it for aesthetic reasons as well as the degradation in sound quality.
A note on the Keyboard/mouse: you might want to reconsider placing them in the trunk. look into buying a Chicony IR wireless keyboard off of www.pricewatch.com because they only run $27. either that, or run the KB/M wires to your backseat. trust me--its gonna get REAL annoying if you have to stop the car to get to the computer.