If the car runs better with the MAF unplugged, then you have a bad MAF or a massive air leak in the tube from the MAF to the throttle body. When the car is cold, it does not really use the MAF, so that would explain why the problem is not so bad when the car is cold.
Yes, the MAF and the oxygen sensor do work together to tell the fuel control computer what to do with injector duration to meter the fuel in at all speeds above idle.
The cranking problem almost has to be completely independent of the bogging/hesitating problem. That does sound like a bad starter if bumping the car to a different engine position caused it to crank.