Your screen size should be based upon your viewing distance and your theater seating preference. From there, you have to pick a projector which will work at the distance you want to mount it, or you have to move the position you want to mount it to a position where it fits your screen size.
So, take your viewing distance in feet and multiply it by 8 inches, and that is a 'center of theater' screen width. Typical would be a 12' viewing distance, and a 96" wide screen which is a 110" diagonal.
Now, you need to find projectors which can hit a 110" diagonal from the throw distance you want.
On top of this, you have to accept that you aren't setting this up in a home theater. This is your family room. Likely light walls, white ceiling, perhaps hardwood flooring, and (worst of all) windows. You aren't trying to control light in any clean manner and you aren't handling reflections, so you should be working with living room friendly projectors like the HD25 or the BenQ W1070. But, if you demand the throw distance you are after, then this may force you into a more expensive model, which may be more theater friendly then family room friendly.
This will leave you with a poor situation during daytime viewing, but will enhance the after dark viewing.
To be fair, daytime viewing is going to be poor if you aren't eliminating ambient light properly. If your walls are white, then during the daytime, pull your shades and look at your walls - that's the shade that 'black' will be when projecting, and white needs to be far brighter.
Generally, a 10:1 contrast ratio may be about as good as you will get in that type of situation. This may be fine for sports, but won't be good for video gaming or movie watching during the daytime. Without light metering being done, it's impossible to truly tell what your real world contrast ratio expectation will be.
With a 120" diagonal and 17.5' throw distance, here is the list of projectors, sorted by price, with 3D/1080p which can support that setup...