Maybe a stupid question, but have you installed the HDMI adapter that connects the Raspberry HDMI to the HDMI of the Game Hat? You probably had to remove it to install the Raspberry to the Game Hat, but after doing so you need to put it back!
For wireless controllers, take a look at the Controllers section in the Docs, you can use one of the PS3/PS4 controllers, an Xbox controller, 8Bit do controllers, etc.
