With the questions you're asking, you may well be in the wrong subforum, these questions lead to answers that I think point you to the $20k+ forum.
Originally Posted by projectornoob
A projector that is rated to project it's image on a 50''-200'' screen. I'd like to increase this greatly (while maintaining the image quality/contrast/brightness if possible).
First things first, the size ratings of projectors are absolutely worthless in determining usability in practice. They're not based on any image quality metrics, probably just physical optical design parameters (like maybe focus range/zoom range or something).
|1: I could just move the projector back and focus it properly?
2: I would lose brightness and contrast in doing this?
3: 2 could be corrected with a different lens(maybe only fixing the brightness or contrast, and altering the throw ratio)?
Projectors output a "fixed" amount of light, the bigger the image, the larger the area that light is spread over, and the result is the dimmer the image. There's no way around this. All else equal, a larger image is dimmer than a smaller one. The only thing you can do to compensate is to throw more light at the screen.
There are several options, in some cases you can open up a iris (which restricts light/improves contrast), run the lamp in a higher mode. But these are somewhat limited in their capabilities. If you're talking very large screens, generally you're looking at machines with "bigger" (more powerful) lamps or multiple lamps.
The other option is a screen with gain (focuses the light more back to the viewer).
|4: Are there any non theater (4kx2k <$50000) projectors I should be looking at, that are still capable of 1080p (or better) and 3d(120hz)?
There's plenty of 3D 1080p projectors, JVC released a number, Sony, Sharp all have 3D 1080p machines.
|5: Does the projector lamp have anything to do with throw distance, or only brightness/contrast?
Just the brighteness (well and gamut), that's it.
But at greater than 200" and 3D, I think you're solidly in the $20k+ forum. I doubt you'll get the horsepower otherwise.