Well, first off HDMI will only push up to 60fps. Mini to hdmi is still ending in hdmi. If you have a monitor that does higher than 60fps you need to use a mini to mini or mini to display port as right now display port is the only connector that can push higher than 60. It can do 144 up to 2k and 4K at 60. If you have a TV it will only get 60 from external connections. This is because TVs lie. A tv may say it can do 120/240 or even 480, but it's just using advanced software to predict the next frame. Sure it may be showing something at 480 but it's the software making that happen. The highest FPS you can get is 144 from a monitor that says it can do 144hz Over display port. HDMI 2.0 I think can do 4K at 60, so if your getting that your all set.
As for your port not working, I would call asus