HDMI can be used for 1080p 240 Hz if the monitor supports it.240Hz is only supported on DisplayPort cables, I'm assuming one of those came with the screen. DisplayPort cables only connect to DisplayPort video outs/ins.
You should get 240 FPS in certain games. Don't expect heavy games like GTA V to give you that kind of FPS though, as heavy games are almost impossible to get up to that kind of FPS. Lighter games like LoL, CS:GO, WoW and others should run on 240 FPS without a sweat. It's mostly only fast paced games that will give you the benefit of 240Hz screens anyway.