A:AnswerYes it has an hdmi on it. I thought I had ghosting issues with my controller but it turned out that since I leave my windows open dust got into the control sticks on my DS4. So be sure to dust your controller if you have the same issue.
A:AnswerI don't believe you can get the max resolution over HDMI. You will need the DisplayPort adapter. I always use DP if available and I have no issues achieving the maximum resolution possible. Not sure how it works for a Mac, as I never use them.