1. reverse camera issue:
the easiest solutions are these:
1a. a relay-- this would prevent small voltages from tripping the relay, which would prevent the screen from switching. shifting between gears could still cause it to switch though-- as you usually need to cycle past reverse to get to drive(if you drive a stick, then ignore this)..
1b. a delay, like the 528t pulse timer might be the best option-- assuming that it is only a small voltage that passes through the reverse wire(small enough to not trigger a relay). setting the timer for about a 1 second delay would prevent gearshifts from triggering the switch, while still working correctly when you stay in reverse. this would add 1second to the amount of time it would take to see the reverse image though..
those are are symptoms of a ground loop, or dirty power.
1. you said that your using the cigarette lighter adapter for power. these are notorious for having poor grounds and bad power, and is the exact reason why so many here recommend running a new dedicated power and ground wire. so this would be the first, and best step to removing your noise issues.
2. you never mentioned how/where the camera is gettting power from, or where it is grounded. having the camera grounded somewhere else, or getting power from a separate wire then the carpc/screen can also cause noise issues.
3. you also never mentioned how the screen is powered-- same as the camera, getting power/ground from other wiring can cause noise.