Just make sure it's high enough to be able to access that area. A injured chicken could be a chore to get if it chooses to take refuge under there, or becomes broody.
Yes, but the OP has limited space. 14x20 is it. With the coop in the corner, OP can have a door to the coop and run, and not have to access the coop through the run
I'd build the coop in the corner of the 14x20 and make the run encompass the rest of the 14x20. You could put the door to the coop on the outside, and have a separate door to the run.
I understand you need room behind a coop to open doors. If you have a small yard I can see how separate coop and run will reduce yard space, but a coop inside a run is going to reduce the runs SQF.
In your situation, it might make more sense. Hard for us to tell without seeing the space ourselves.