The arm is keyed to the square edges of the motor shaft. You'll have to wiggle it a bit before it will slide down on the shaft securely. Sometimes, just tightening the bolt won't be enough to get it pressed on the shaft if its not lined up correctly.
Once its on the shaft and pressed down, it should be fine.
You might also check the angle of the stop tab and make sure it matches the other side of the car. I had the same thing happen to me and once the arm started going by the tab, it had leverage on the very end and bent it a little.
Here's a good tip for replacing the plastic bumpers that you are missing: Looking for Firebird headlight door actuator bumpers