as a guy who does live video manipulation for events (concerts EDM shows) And as a guy who has done full dome (planetarium) video work, I've got to say I am envious of that room. If i had that room in my place I'd have it tricked out like you would not believe!
First thing.. you didn't provide dimensions of your room, as in projector placement options? Figuring out your throw ratio is where to start, and then you can get some idea of what projector(s) to use for this. Also how much area do you want to cover then entire thing floor to ceiling? how about the ceiling, that to? What kind of things do you see yourself projecting, movies, or are you thinking ambient video to use like decor? how important is resolution to you?
with a budget of $3000 you'll have to keep your expectations realistic. A used Panasonic PT-D5500 (5000 lumen) projector will set you back about $1500, and a 0.8:1 wide angle will set you back another $1000, but is that enough throw? if you need a shorter throw then you'll need to break the signal up into multiple ultra short throw projectors, and use software to edge blend it all together. There is some really decent priced multi display software available these days, and it's fairly easy to use.
here are some links for you too check out