I run a 6870 with a similar CPU and it will push most games to full graphics at 1080p. Anything super GPU heavy, such as Battlefield 3 or Metro 2033, will require lower graphics settings to achieve nice frame rates though.
As far as Dayz, I had never heard of that until I Googled it a few seconds ago, so I have no idea. You should be fine though, based on how the 6870 performs in other games.
assuming the psu can at least hold half of its specified wattage, you shouldnt have trouble with a 6870. an fx-4100 and a 6870 is fine(its similar to phenom 965be + 6870 to some extent) and will do okay, except that arma2 itself is a badly coded game at max resolutions. the build will do fine on smaller resolutions under 1920x1080 I belive.
edit: though IMO, moving from a 6770 to a 6870 isn't the biggest of jumps. i would look into a 560ti or a 6950(when its on 200$ sale)
The motherboard should be fine, I wouldn't worry about it.
And yes, a 6770 will play most games on at least medium detail. 1080p may be out of range though, you might need to drop down to 720p. Again though, i don't know how demanding Dayz is.