There should be no issue in using the 1/2" thickness. The reason VanillaGorilla mentioned three-quarters is because that's the usual total thickness (including plexi) people go for in order to use the most common size of t-molding. There are other sizes of course but 3/4" is the most common.
3/4" will be a little stronger but that should be irrelevant really unless you're planning on really giving this panel a beating. Bottom mounting the sticks is usually the way I go too because I don't like to have long shafts. I prefer my sticks to be shorter and slightly stubby like the classic sticks of old. But that's just personal preference. Each to their own.
I'd suggest 3/4" mainly for the joystick support. I wouldn't want to bottom mount joysticks because then the shaft barely sticks through the panel, and it looks and feels wrong. To top mount the joysticks, you need to route ~ 1/4" bed on top for the mounting plate to rest in (and maybe more because I never found perfectly flush mounting screw heads). If you go for a 1/2" panel, that only leave you 1/4" for the screw/nut to grasp. The screw holes are already in a precarious position, very close to the joystick body, so I think that a 1/2" panel won't provide enough stability for the joysticks.
Also for Ultimarc trackball, the mounting screws supplied stick out exactly 1/2". If that's the thickness of your panel, the top of the screw will just peek through and be exposed on the top of your panel (may not be a problem if you plan to cover it with some other material.