For the door banging over bumps, if none of the above easy fixes cure it, there is one harder fix that I stumbled upon by accident. When I was repairing the frame on a 80 f150 I had, i was replacing the crossmember that the rear of the cab sits on. I found a good rust free rear frame section from the junkyard from a later truck, brought it home, and noticed that under the large rubber cab mount, the crossmember had a nice round rusted out hole right under the rubber. When I lifted the cab up on my truck, sure enough it was the same way, completely rusted out, but it looked like a clean round hole right under the mount. Once I patched the newer crossmember and installed it, that's when I noticed the door banging i used to have was gone. Stands to reason, since the whole back corner of the cab was now supported by the frame.
I am thinking if you crawled under the truck with a crowbar, and tried to pry up on the back corner of the cab, and if it moves up and down, you have the same problem.