A free projector of decent home theater quality is certain a good starting point for your budget. Pairing it with a grey screen may be of some use, and at 106" diagonal, you should have plenty of light output to do well in there. While the Optoma isn't as nice as the BenQ short throw model, it is still well received and does a good job.
Unless you intend to paint the walls dark, I would likely go with a grey screen to improve black levels with that model. They still won't be perfect, but a .8 gain (or so) grey screen will help.
The screen size/viewing distance looks appropriate. About 10" of diagonal for each foot of viewing distance is common practice. You are pretty much right on.
Two things to consider at some point: Lighting and wiring.
Wiring I've already gone over, so it's in your head. Also be aware that with front projection, there is absolutely no reason to have all your equipment (and flashing lights) at the front of the room. They can be hidden off to the side, or back of the room to keep things out of the way and put all your attention on screen.
For lighting, you can really have a lot of ambient light on while still enjoying the game on TV, you just don't want any flood lights on. That is, non-directional lights that just fill a space with light will significantly impact on-screen quality, while directional spot lights will give you lights to move around, read, eat, etc. with minimal on-screen impact. See:
http://www.avintegrated.com/lighting.html for examples.