If your cabinet sides are structural (most are), you'll probably want 5/8" or 3/4" panels. The CP can be made out of something thinner, since it doesn't bear a lot of weight. You might have to rout out a recess for the joysticks no matter what thickness of wood you use, because some joysticks are designed to be mounted in a metal CP. The best bet is to mount one in a scrap piece and see if you like how far the stick protrudes from the panel. It might be too short, in which case you can decide whether to mount it in a recess on the top or bottom of the panel.
You pretty much covered the plywood vs. MDF debate.

MDO is great stuff -- it has the advantages of plywood (holds fasteners better) while giving you the great paintability of MDF. It's pretty easy to work with, but like regular plywood, doesn't take an edge well. (I.e., routing the edge is not going to give you particularly nice results.)
Properly prepared MDF or MDO gives you the best painted finish with the least amount of agony. You'll go through a lot of sandpaper and paint trying to achieve the same finish with regular plywood. If you do decide to use plywood, get one with a sanded finish on one side (birch is the most common) to save time and effort.
T-molding slots are easy to make in MDF with a router and slot-cutting bit. Harder in plywood because of tearout, as you mention.
Regards,
-- Chris