Its a problem with windows and the PS4 controller. The driver uses Direct X's DirectInput , while this game uses the newer XInput.

http://forums.pcsx2.net/Thread-DS4-To-XInput-Wrapper will help fix the button problem. It will create a virtual Xbox360 Controller in your system, but it will really be the PS4 controller. Set the game to use this controller.

I don't know how it will work with the PS4 Trackpad though