I have been looking at TVs for like two months now. What ATM said is spot on. Something like this is pretty good (far outside your budget I know).
http://www.amazon.com/gp/product/B00TWFHELW?tag=rtings-tv-pm10b-20&ie=UTF8
The input lag is about 27ms, which puts you about 1-1.5 frames behind a gaming monitor at 60hz. You can go cheap on TVs, the Vizio E series is a good example, but your input lag will still be an issue. Further, if you cheap out on a TV you won't have 4:4:4 chroma and then using your TV as a workstation will just suck.
I really think you are better off staying with monitors, on the whole, because your budget doesn't allow for much else. I am willing to spend 2k on a TV, four times your budget, and I still come up empty. Really, the best you can do at the moment is the Samsung 7100, the Vizio E series, or the Sony W850B (for split uses - gaming / TV). But realize, and maybe it is because I come from plasma TVs, these new LED TVs can have and usually have lethal problems. Half the time - moreso than that for me personally, anecdotal evidence ftw - whether you get a good one is pure luck. I'm on return #5 to Best Buy because of unacceptable clouding, flash-lighting, radial banding etc., none of which existed on my plasma.
It may not seem like 15-20ms is a big difference, but it can be depending on the game. It isn't so much whether you notice it, its more or less that no matter what, someone else is that far ahead of you. Again, depends on the game and the skill level you play at, but seriously, I can only encourage you to stick with monitors unless you are doing single person games. Then, it doesn't matter. Anything below about 35-40ms will be unnoticeable to you in those scenarios.
If vsync is turned off, will that get rid of mouse input lag on a TV?
No. Although having it off means less lag because enabling v-sync adds lag.
Pretty much all decent TVs nowadays have "game mode" or similar and the same applies to AV receivers. (For the latter, it's generally the cheaper ones that have less latency!)
Which, depending on your gaming needs, will still put you behind real gaming monitors by 15 or so ms (at least). Again, may not seem like much, and isn't for most, but it could be and is worth knowing ahead of time.
It's hard to notice with a controller, but it's there.
Most console controllers have input lag of their own, somewhere in the neighborhood of 40ms. It all adds up. No point adding even more via your display.
TLDR: Stick with the advice given already. And let me add one bit...increase your budget / save. Why spend all that on display power only to gimp the end result?
PS ATM, school is over in two weeks and I only have my 4690k to 4.4 @ 1.25, so it's a dog. Gonna harass you when I have time so we can get this guy up to 4.8ish. Haven't breached 53 degrees C yet with Aida, so I have some head room.
