1. Is this a common problem?
It's not unusual, not sure I would call it common
5. Can you get better wireless service depending on your location within the hotel (eg lower floors)?
My profile asks for a room far away from the elevators and that usually means the end of the hall.
That also means usually lousy internet service for wireless. It all depends on where the routers are, but generally if they offer wireless in the lobby it's full strength. You could call and ask, I mentioned it in one place the signal didn't reach my room (not to complain, just to mention it in case they had a router that was down), and they pulled out a sheet and showed my room highlighted as poor, with some others as good and fair. So they possibly know when you are checking in.