1. Take a look at the Intel Atom mini-itx board. It will do what you want. They're available from many places but if you like what you've read here, buy it from the mp3car store and help support the forums. FYI - the budget bundle on the store will probably do all you want and includes an integrated power supply and hard drive (add $235 for the SSD drive). Just wire it to the car, plug in the 1/8 output cable and connect the monitor and you're done.
2. A gig of memory will likely work just fine.
3. If your harness accepts regular line inputs like from an mp3 player, it will work as you described. Test it with an mp3 player like an iPod and you'll see.