I'd like to eventually get one as well, but honestly they aren't great for gaming, especially FPS. If you plan to play online, prepare to be pwnd. You'd think that a big screen would be great for FPS because everything would be easier to see, and it might be for some people, but for most it's just a pain in the ass. You can target easier on a large screen, but noticing someone pop up from behind a wall in your peripherals don't work as well when you have a bigass screen...
If you do buy one, make sure you check the NATIVE resolution. One of the common marketing tricks is to advertise something as a "1080p projector" because it accepts 1080p signals...but in the technical specifications you'll see the resolution is shitty and clearly not HD, and in actuality the projector will downscale your crispy video down to a shitty 800x600 or 1024x768 resolution. If a projector with 1280x720 or 1920x1080 native resolution is not in your budget, I honestly wouldn't bother. Personally, I wouldn't bother one without 1080p, as I don't like to see large pixels
I know a company or two released the first sub-$1000 1080p projector last year with decent contrast/brightness, but sorry to say I can't remember the name.
Also realize that it will not be very friendly on your wallet to use as a daily thing. The bulbs burn out, and are expensive to replace.
Good luck finding one with a coax input. Those are pretty old school and only n00bs use them. Most have at least composite inputs though. Really though, why buy a sweet high-def projector and then send it a shitty signal through a shitty low-res cable? Be a man and use component or HDMI. All cable/sat providers have receivers with HD outputs now (obviously)...
For the screen, you can buy cheap screens on ebay that will work. Google "projector screen paint" and use that. They usually have somewhat of a silverish color to increase contrast, brightness, and clarity. White MATTE paint will work fine, though.
Not all projectors need a pitch black room...just most of them, and probably all of the ones in your budget. Also realize that the bigger the screen, the farther back it will be, and the darker the image. Twice the distance = half the brightness.
I would like a projector and plan to eventually get one, however it would be a secondary screen and I'd never recommend it is a primary display... Bulbs are pricey, it's best in a dark room, not the best for gaming, quality is easily surpassed by TVs (unless you buy the high end, $25k+ projectors), and they suck a lot of power.
If you don't have a nice TV already, get a big DLP. They still have bulbs you'll need to replace, however I think they last longer. The picture quality will be significantly better, as they have great color reproduction and contrast. For the same price as a decent 1080p projector, you can get a huge 3D-ready 1080p DLP.