You can just solder the wire directly to the laptop. Turning it on/off by ignition is tricky, you will need a shutdown controller for this. I have designed one but havent done a full test on it.
Check if your laptop support wake-on-ring serial port. If this is so then you can use that to turn your PC on. All you have to do is emulate a modem "ringing". I havent done any testing on this one though...but it should work ok.
Heres the one I done for shutdown thought you might need it, as posted on another forum :
COM..|..............CIRCUIT...............|..Ignit on Switch
.............1N4148.....1.2K Resistor
Pin 1 ------|<|-------/\/\/\/\/------> Connect to + 12V, "Low Battery"
.............1N4148.....1.2K Resistor
Pin 8 ------|<|-------/\/\/\/\/------> Connect to +12 V, "PSU failed"
Pin 5 -----------------------------> Ground
***The inputs are configured as negative triggered on the UPS control panel***
The resistors are there to limit the current to 10mA. The diodes maybe required to remove any stray voltage from other circuit of the car that may trigger the serial port to a HIGH state.
"Low battery" input = The UPS battery is running low
"PSU failed" = The PSU has failed and PC is running on battery.
Its up to you how you want to wire the 2 inputs to the car. You can instruct it so that you have "Failed PSU" when your battery is not being charged by the altenator. Then "LOW battery" when you turn off the ignition.
NOTE : You must configure the UPS setting on your control panel correctly so that the input requires a negative input.
DISPLAY PROPERTIES > SCREEN SAVER > POWER > UPS
Select a GENERIC UPS and customise it :
Power Fail / On Battery = NEGATIVE
Low Battery = NEGATIVE
UPS Shutdown = Un-tick it as you dont have any real UPS to shutdown
Then configure your UPS :
Just make sure the next step is SHUTDOWN PC. You can un-tick all the notifications as you dont really need it.
VGA-to-TV coverter is available all over do a search on google. The picture quality wont be good as you be using a video monitor...and NOT VGA monitor.
Hope that helps