the only thing remotely related to what you are describing would be the dome light powering something through a bad ground.

Anything that would make the car crank wouldn't be related to that?
The most i could see it benefiting you would be priming the fuel pump.. again not likely.

before you stick a starter in there, try bypassing the starter relay, check the ground and Verify the neutral safety position. Not just with the shifter, but the wiring.