Sleeping Dogs is similar to GTA, but relies a lot on martial arts instead of ranged weapons - it's different, but it works pretty well. I consider the driving a little weird, and the cities are really crowded... but it's a neat game.
Deus Ex is one of my favorite franchises, but of course Human Revolution is not without its quirks. Reviewers were mostly unanimous in saying it's a great game with good atmosphere, but the boss fights were out of place (which I agree with). I think the stealth sequences are almost more arcadey (rolling between crates) than it should be, too. I didn't think it was short, but I take my time.
Those are two of the supposedly really good games of the last few years. I've heard good things about Far Cry 3 Blood Dragon as well, and that one was intended to be DLC/relatively short, which they made standalone. The core Far Cry 3 game is supposedly one of the best open world shooters, but that's not included in Blood Dragon.
Since you mentioned you have kids, you might want to factor in the game's content. In that case I would say that Sleeping Dogs and Deus Ex are pretty much the same with respect to violence. I'd probably say Deus Ex is more kid friendly. FC3 Blood Dragon is an over-the-top 80's shooter, so obviously it'd be a little more aggressive.
I'm not huge into racing games, so can't comment on GRID, but it's almost definitely the most kid-friendly title listed - and sniper Elite doesn't strike me as the kind of game I'd like either... just b/c of its placement.
Try to check out metacritic and/or ign or gamespot for some reviews, or watch short lets play/reviews on youtube to help get a better feel for which you might like! All of those sources should help you make a good, more informed decision as well.
The 7770 should be capable of handling those games at 1080p just fine... it's the card I use and I didn't run into any problems with Sleeping Dogs or Deus Ex, at any rate.
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