I can save you some time from the projector side of things (if this question is still valid and you're considering one):
Not much of a projector person myself but about 3 months ago I got bit by the projector bug since I'm busy redoing my entire living room and wanted something that's much larger than the usual 55" TV's you get in the R10K price range (and I despise the extra money you're almost forced to pay these days for "SMART" TV's when all I actually want is the looks and panel quality found on SMART TV's... good luck finding that)
Anyway so without a doubt the best projector you can currently buy when you're on a tight budget is this one:
ViewSonic PJD7820HD Full HD 1080P Native with HDMI, 120Hz Active-3D Projector
http://www.wootware.co.za/viewsonic...ga-composite-and-s-video-black-projector.html
I'm not sure if some other competitor has yet released something competitive for the price/performance you get, but I doubt it.
I know R10K is a bit above your budget but I'd rather save up the difference, IF you're going to want to use this thing for web browsing etc. as well. Also, for once the lumen specs on this thing isn't just pulled out of some marketing person's arse, in fact, it actually surpasses the lumen ratings in real-world testing.
Gaming on this projector is pure pleasure. Latency is extremely low (from what I've researched it sometimes surpasses CRT latency even). It's not a cinematic projector in the true sense which means you'll get unnatural colours etc. if you don't town down the brightness a bit on the movie side of things... luckily I'm not a videophile with my finger up my arse so this doesn't matter and I can't even notice any issues after doing some basic calibration... the higher contrast means games look awesome and I was surprised to find out just how much more enjoyable dark games (Splinter Cell, Metro, Dead Space) are with something like this... probably because of the lower blacks etc.
I don't even have it projected against a screen yet, just a relatively freshly painted darker grey wall so I'm excited to see what difference a screen would make. People who are more used to projectors will probably go on-and-on about how incredibly bright it is... for someone like me though who doesn't really like projectors the brightness is just good enough.
It's still visible enough even with direct sunlight streaming into the room (though you wouldn't want to game that way, unless it's something like Mortal Kombat or other less intense 3D games).