I suggest the Viewsonic Pro7827HD, it’s within your price. Not really short throw but can go up to 100” in that confined space. 100” is a lot, believe me. You won’t want to use all the available wall space, anyway. It’s good to have some space away from the walls and ceiling. You’ll have to mount it on the ceiling or a low shelf or table. This projector has audio output, and you can probably connect it to your system using an audio to RCA adapter.
About the pull down screen to cover the door, that’s exactly what I did in my space. Except that I built my own screen. It is true that pull down screens tend to get wavy, but I’ve seen a few in this forum that looked pretty good. On the other hand, if I had a good straight and smooth wall, I wouldn’t hesitate for a second to use it. So much more practical than using a screen.
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