While I've never built an arcade cabinet out of Melamine, I've built numerous storage cabinets in my basement out of it.
1) Special screws. Yes, there are special screws that have a double thread, one wide and deep and another interlaced that is not as wide. They also have no thread the first 1/2" to 3/4" and have a pure flat head to keep from needing to countersink holes. Also they are 2 to 2 1/2" deep. My local "Fleet Farm" has a line of shelving products I believe called "EZ Track" and has the best deal on parts. You get a bag of 16 shelf pegs -- the good ones with the "spoon" to hold the shelf not just the round peg, and then 8 screws and 8 white screw caps. All for $1.79. Most stores want that much for JUST the caps or JUST the screws or JUST the pegs for that many. If you don't want to use the finishing caps, you can screw in a little tighter and then use Elemers white colored wood filler to seal over the top and then touch up with a semi-gloss white paint. No, it's not perfect but works.
Note: I've noticed a significant decrease in the quality of materials for standard "sheetrock screws" in the last 4 years. They used to have a very thick taper from the head into the screw. Now the taper is about 50% narrower than it used to be and I've broken off a large number of heads by using them to attach things other than sheetrock, and that's including hand screwing, not just power screwdrivers. I would strongly advise NOT using sheetrock screws on Melaine. Also they are not going to be long enough to provide enough support and risk breakage on the cross piece. Go with at least 2" screws, no more than 2.5". If you must use standard sheetrock screws make sure the head taper is very beefy so you wont twist the head off while you're screwing into the dense particle board. (If you break it, you're screwed, no pun intended.. you wont be able to get the broken part out and have to drill another hole and use another screw)
2) Chipping: Yes.. but not with a jigswaw, much anyway. here are tips you can use to prevent chipping.
- Jigsaw: Use the finest tooth blade you can find
- Circular, table or Miter saw: Precut a channel into each side where you will cut using a carbide tipped blade. Only cut into the center of the piece, not all the way through. Drag the saw so the blades cut INTO the top, not spinning away from the movement. Flip and do again on the other side, then adjust the blade and cut all the way through. It's the blade attempting to go through both sides and moving in the wrong direction that causes chipping on the side opposite of your cut.
- All: You can possibly reduce some chipping by using masking tape on the cut line on both sides. Press very firmly to make it adhere to all the surrounding material.
3) Countersinking: See #1 However, for my finished room facing ends, I buy 5/16" wide solid wood dowels and then peg those sides and adhere using PL400 (Polyurathane based plywood glue in a caulk tube--strong than the glue that holds plywood together). use lots of pipe clamps and let it set overnight for a strong seal. That way nothing is exposed. However involves two problems; Possible breakout if you over hammer the side on (use a large rubber mallet with a large solid board over the peg spots to reduce the risk), and also not drilling the holes in the side and cross piece exactly so things don't line up perfectly. But this way prevents any holes in the outside of your cabinet.
4) Yes, melamine will sand or plane well. I suggest C-clamping both sides together into a sandwich, then using a hand planer to work out the major high spots, THEN take a random orbit sander with a low grit (40-80) paper and work everything flush, then take a 120 or so to finish it off. You'll waste too much paper using 120 to start with.. Plane it first. Hand planers are about $5 at most stores for the standard Stanley ones. And get a circular random orbital sander. They cut soooo much time and materrials off sanding things. They work great. They're about $25 on sale at the right stores, $50 if you just walk into an expensive store.