What Arduino Board would I need?
I would go for the new Leonardo because you have the USB functionality in the main processor. With an UNO you have to program the main processor as well as the ATmega16U2 that's doing the USB protocol handling.
Would I be able to configure the Analog stick after connecting it to the PC? Like mapping keyboard buttons to the Stick: Up=W, Down=S, Left=A, Right=D. The analog function of the Stick is not needed.
That's done on the PC if you connect the stick as a joystick. I'm sure you'll find software which is able to do that. The other possibility is do emulate a keyboard from the arduino and translate the joystick movements also there. Take a look at the keyboard example of the Leonardo.