I just bought a new monitor, it's a HP 27xi, I also have a Sapphire Radeon r9 270X and there is a black border around my monitor at 1080p, also no sound on it
Sounds like you are connecting through HDMI without the proper drivers. the black border can be fixed in catalyst control center in the GPU scaling options. The sound should be the HDMI driver.
I fixed the black border but the no sound remains, i plugged in a pair of ipod headphones and it worked, but no sound on my logitech g930s
does the logitech connect the same the ipod headphones do? 3.5 mm audio jack? are you using the SAME plug in for the ipod and the logitech? sounds like a problem with the headphones then. maybe it needs super special drivers