1. Most inverters cut out at 11 volts. So with most inverters, yes the computer will reboot if you have the computer on before the engine. The ATX DC-DC Power supply I built will supply its full rated current all the way down to 8 volts. Some other DC-DC power supplies will go that low too. My car's battery drops as low as 8 volts during cranking. Most new batteries drop as low as 9 or 10 volts.
2. Most AC-DC or AC-AC power supplies are about 80 percent efficient, most inverters are 85-90 percent efficient. If you use both, you get the idea...
3. I'll sell you one, but cheep to me is about 80 bucks, and that is how much all the parts alone cost!!
4. If you use a celeron, 566-800, a dvd, gps, standard 5400rpm or 7200rpm hard drive, and an all in one motherboard, 70 watts should be just fine.