If you're using an original Pi B a lot of your issues may be solved by upgrading to the Pi 3, it's considerably faster which helps to smooth a lot of the grumpiness out
Could you send me a picture of the stick you're looking at please? Most of the joysticks I've come across are 5 pin Sanwa or 8 pin (2 x 4) HAPP, so yours may be the latter. It might also be a custom unit built specifically for the Hori box
If you check the image below, next to the 5 pin connectors for the Sanwa joystick (top centre) there's a line of 4 x 2 inputs (top right), this is for HAPP style units.
No reason why your Pi won't be happy uncased as long as it doesn't get bashed around during intense fighting games!