I wouldn't worry about the thickness too much. My coinco door just uses carriage bolts and nuts to mount on the cab. My Happ over/under uses a screw down bracket that just clamps onto the cab. Worse case scenario is you cut a 1/4" piece of plywood or something to use as a spacer. Weight shouldn't be a problem on the 1/2" MDF.
The Midway doors are nice, but if you have the room you're probably better off sticking to a standard type door. It'll stick into the cabinet further, but on an upright that rarely matters. You can probably find one cheaper than trying to find a Midway in decent shape with all the necessary parts. They're not rediculously heavy, but being solid steel I'd definitely check shipping costs. Check the B/S/T forum and then hit eBay. You also might try to make a few calls in your area to see if any local operators have any to spare. You should be able to find one on eBay for less than $20 with shipping if you're patient enough.
On a standard one-door coin door there are a few main parts:
The door itself contains the outside coin slots, coin return chutes, and switches that the coin hits as it passes through.
The frame that it attaches (via piano hinge) which mounts to the cab.
The coin mechs that attach to the coin door. They connect the area at the top where the coin goes in the slot and the switch that it hits on the way to the coin bucket. It also sorts out if it's the right type of coin and sends bad ones to the coin return. These are fairly universal and fit most coin doors. They are also available for different types of coins. So if you want to use .984" tokens, then you'll need the coin mech that is made for that.
Lock. You can replace it with a "mailbox lock" from Lowe's/HD/any hardware store/locksmith. Standard cam locks use a slightly bigger body that won't fit in the hole.
You'll also usually have Lights, wiring, and maybe a coin counter as parts.