I have the generator under the bed, it's fine for our use, we don't can't where we need the generator running at night. I will say it was quieter than I thought it would be. I am going to add some sound material around the box that it's in.
For the rear door/screen. No you don't want the AC running while you have the patio open if all you have is the screen. The AC would work so hard, it would freeze. However, some have a 3 season door. I don't have one of those, maybe someone else can comment as to the effectiveness of the 3 season door, but they would hold the cold in better than a screen.