Intel HD4000 you mean? I use my 2012 MBA 13" (128GB/4GB) for casual gaming, including War Z and Borderlands 2, which on reduced graphics settings and lower-than-native resolutions does the job. It's definitely not a "gaming machine" though.
I think you mean HD 4000, or maybe the Haswell 5000 which I guess you think will be on the 2013 machines. In that case, we have to wait in see. In the former case, ok at low settings and resolution.
Only time will tell. There's not even leaked performance benchmarks out yet, so nothing's certain. With settings on "Medium/Low" rather than "High", it might be able to run 1080p at playable rates, but it remains an ultraportable so you definitely shouldn't expect it to be ultra-smooth.
As a matter of fact it's speculated that for the ultrabooks performance would be +50%. Good, but not incredible. 1080p might be a no-go. Also, I suppose you are talking about running games on Windows through Boot Camp? Performance for games is better through DirectX on Windows than on OS X.
There is no technical reason why it wouldn't, besides DayZ being one of the most poorly coded games known to the industry. Even a desktop GTX 680 has trouble delivering 60 fps at 1080p, I don't think you will be happy at how an IGP will perform.