In my experience the best type of monitor to use is a GSync, but not everyone uses NVidia, or spends the coin on a GSync monitor. I have tested three different GSync monitors and none of them use HDMI in support of GSync, though one of them actually had an HDMI port it did not support GSync through that port. The two that are supported are Dual-Link DVI, and Displayport. So, Displayport would be my suggestion.
Audio across an HDMI cable (through your video card) is going to lag and lose sync as the number of 'voices' increases. It's as bad as the worst USB, or it can be at least.
Having said this there are lots of people that do not even get 40fps in AH and love their USB audio headsets. I prefer GSync and Sound Blaster ZxR and after experiencing both I must have both.