You can safely remove anything that neither you or the controller need. It's common sense: If it's on there, it's probably for a reason, right?
The little connector and plastic sheet are for the buttons: remove either of those and you can't use the pad's buttons anymore. The black thing with all the silver legs on it? That's the chip - remove that and ... well, you can probably guess how bad the result would be.
If the pad has a chip, keep it. Resistors? Keep 'em. PCB and analogue sticks? Junk 'em, you won't need 'em for your GH mod, right? Just work your way through the list, keeping in mind one final thing:
If you're gonna cut up the PCB, and it sounds like you are, you'll need to keep the important bits, like the little circuit traces that connect all the components. If you cut, for example, the trace providing power to the chip, it won't work anymore. If you really need to make it smaller you'll end up replacing the PCB's circuits you cut with wires so the whole thing keeps working.
It's not rocket science.