The 7770 is better if gaming is the sole concern, but it isn't in this case. It's worth giving up a little gaming performance to have good Linux compatibility IMHO. An Nvidia card also opens the door to experimentation with CUDA programming, which is an awesome skill set to have. Regarding mini-HDMI to HDMI, the only difference is physical; they are are electrically identical and nothing will be able to tell the difference between the two. An adapter or mini-HDMI to HDMI cable will work fine.