Fredrik - Welcome to the forums. My first suggestion would be to start a thread in the project announcements so that you have a greater chance of most everyone helping with you with your project.
Second. Let me answer some of the questions that you posed for us.
1. I think everyone tends to use MDF because it more readily available and easier to work with. Over here 9/16th MDF is not available but you can use something close. Like say 5/8th's. If you can find 9/16th plywood go with it. It was the original size of thickness on the sides and you can easily find what is called Nintendo Flat T molding when you go to choose your t molding.
2. If you want to build a DK cab I would stick with the DK plans. If I remember correctly DK Jr is slightly different. I have a copy of the Jakobud plans if you need them. Just PM your email and I can send them to you. Keep in mind after doing research for my FIF Jr cab (which is just a DK cab) I found that jakobud himself said that the plans he made are not exact.
3. Like everyone has stated use a 4:3 monitor. It will be easier this way instead of tracking down a true arcade monitor. I'm glad to also hear that you are going with MAME. Because if you went with an LCD monitor and got a DK PCB board you would have to also purchase all the other internal parts to the arcade and also get a certain adapter for the monitor to hook up to the PCB.
4. The coin door for me was sort of a pain but I kept on ebay, KLOV and the buy sell forum on here and eventually found one that I had to fix up a little but came out good as new.
When starting your cab I would definitely agree with Malenko and Jennifer. Collect all your internal parts to get the MAME working. You can do this at the same time as building your cab. So if you run into a road block on one issue of lets say building you can head inside and work on the internals until your issue is resolved. This also comes in handy if you have a poor weather day and you want to hide inside from the elements.

I would also not worry too much about the artwork and parts until they are necessary. As Malenko suggested it would be smart to get the body built then order your artwork. You don't have to go with Mikes Arcade for the artwork so if you are a 1/4 inch off...eh sorry...or a certain mm off, you can adjust the artwork on the computer and send it off to a local printer. Which cannot, even in Sweden, be hard to find.
Let me know. I might be able to get you a high quality version of the artwork. I know if I can't someone on here will have a copy if you ask nicely.

In the end take your time and don't be afraid to make mistakes. I have made two cabs to date and what I mistakes I learned on my first cab I found I didn't make on my next. I also don't want to be this guy because I am not a fan of others posting their cabs in someones build thread but I want to give you the link to my FF Jr cab build. The reason I do this is because I did a lot of research on building it and it is essentially the same cab as a DK cab. It can be found
here.I also wanted to give you the links to a couple builds that helped me along the way but I some how deleted them. Its late here in Michigan so I will look in the morning when my eyes want to cooperate. Happy building Fredrik!