The obvious first question is whether or not this is intended to be a true theater space or not. If it is, then paint the ceiling. If it's not, then consider painting the ceiling anyway.

I just walked int a home that likely spent $50,000 setting up their theater with top shelf gear, in a dedicated room, and they had no lights and white ceilings.

You decide whether you want a great image or to compromise on image quality over $40 in paint.

That said, if you aren't painting the ceiling, a grey, .8 gain screen would be my choice. It will impact the punch of the whites, but will improve black levels overall, which is good.

For better contrast projectors, the LCD and LCoS models are a cut above. The JVC models are easily the best, but the Sony and Epson models are solid performers and any will really do a good job.

You want to use a flat paint, and anything darker than white will be an improvement. The darker the better, but you must work with your wife as your sanity allows.

I used a storm cloud grey in my last home and it definitely was an improvement on white. Dark grey carpet, burgundy walls, and dark grey ceiling.

Haven't figured anything out in my new home.