Hey man, welcome aboard. I'm just starting my cab. I've got everything and construction has just begun (very slowly, I'm no woodworker). You've got some good questions, though.
Melamine: This is an industrial strength particle board that is laminated on both sides. That's right, it's already got a thin formica like laminate covering both sides so you would not paint it....unless you were talking about painting on some side-art or something....otherwise you would just buy it already laminated in a solid color, and once again it is laminated on both sides. Melamine just happens to be what I am using to build my cabinet and I just purchased a few 4'x8' sheets of it which brings up another point. It's heavy. 95 lbs. per sheet. This wouldn't really matter so much if you were just building a joystick and not a whole cabinet, but seriously I'd just use plywood if I were you. It'll be much lighter and much cheaper and much easier to find.
L-brackets will work fine but an alternative would be to use 1"x1" blocks of wood that you stick in the corner where you would otherwise have used an L-bracket and screw both boards into the 1"x1" block. Surprisingly this is actually a stronger method than the metal L-bracket. For an example (and to just learn a hell of a lot of info in one nice place) check out:
www.1uparcade.comHe also used (black) Melamine to construct his cabinet and then laminated over it on the sides and front to make it yellow. Click around his website and you'll see the 1"x1" lenths he used to fasten the side panels and back panels, etc.
As far as cutting Melamine goes what you really need to worry most about is chipping the laminate as the blade enters and exits the wood. There are special blades made especially for cutting melamine for table-saws, circular saws and jigsaws. I haven't cut my wood yet, but I've got blades for my circular and jig saws (the jig-saw ones were like $5 for a pair so I figured I'd get them and see how they do). The best way if you want no chipping is to cut outside your line and then use a router to shave it down to your cut-line. Since a router spins WAY faster than any saw it makes much cleaner cuts. Oh yeah, it was recommended to me to tape the wood with masking tape over the cut line on both sides to supposedly cut down on chipping. It can't hurt.
For lexan most people seem to be using 1/8" to cover their control panels and marquees and 1/4" to cover the monitor area. This comes down largely to personal preference, but those above figures are by far the most common, I think.