You can use almost any 16:9 projector. Don't get suckered in by advertised lumen ratings from manufacturer. Especially with DLP projectors. They tend to be way off.
For the money, on a projector, I might go with something like this:http://www.projectorcentral.com/ViewSonic-PJD7822HDL.htm
The review indicates it can deliver about 3,000 lumens, if needed, put calibrates down to about 1,000 lumens in 'best' mode. So, you do have a range of quality available to you, and it fits your budget, and it is 1920x1080 native resolution (Full HD).
Rear projection will push the screen 10+ feet further into the room, so you will lose that space. You also will need a proper frame for the screen and the material as well as have a slight drop in image quality. This is why front projection is more commonly used, but that's up to you.
The Viewsonic I listed will hit a 120" to 133" diagonal from about 10 to 15 feet away. So, that fits your criteria.
There are a number of DiY rear projection materials, and you can even start with something as simple as a flat bed sheet.
But, Carl's Place is a common stop for the DiY screen maker:http://www.carlofet.com/projector-screen-material/rear-projection-film-projector-screen-material
Worth note that if you intend to build a screen yourself, they tend to bow towards the middle if weaker wood is used, and typically are braced on the back side, which is tough to do when you are rear projecting.
Elite also has some rear projection screens and materials...http://www.elitescreens.com/products/rear-projection-screens
Better yet, they have them on Amazon. 120" for about $230 and 135" for about $280.http://www.amazon.com/Elite-Screens-Yard-Master-120-inch/dp/B00MAPCZEM/ref=sr_1_1?ie=UTF8&qid=1453472445&sr=8-1&keywords=elite+yard+master
I would probably get that Viewsonic...http://www.bhphotovideo.com/bnh/controller/home?O=&sku=1106945&gclid=CjwKEAiAw4e1BRDfi7vghaWU9jESJACzo9jusRjI1Iz_x7Hu3guCC0_Lt_b0cePV-aUoKTCrtrgIwBoCxPXw_wcB&is=REG&ap=y&m=Y&A=details&Q=
Then the screen for $280.
That leaves you $120 for cables and whatever else you may need.
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