I’ll add my voice here.
0.1mm accuracy is certainly possible. It will take some dialing in. I have done some projects that require a very close fit, and have been happy with results cut in materials from foamboard, to hardwood. A circle => 38.875mm and <= 38.925mm is a little easier than a panel => 1038.875mm <= 1038.925mm because there are fewer steps with which to possibly accumulate errors over. As such, I think that as long as you are not displaying Ningi, you’ll be fine. (Sorry, my inner geek got to the keyboard for a second. For those who do not know, the Ningi is a fictional currency from The Hitch Hiker’s Guide to the Galaxy and is a triangular rubber coin six thousand, eight hundred miles to a side. – If you have one, I would be extremely impressed 

)
Letter sized paper, the 3040 is probably more than enough, but 12"x30"? There may be ways to do it, but none of them are going to be easy, certainly not “beginner”
Is a CNC “The Right Tool”? Mostly depends on you. If you’re just looking for something to do this, and realistically won’t be doing much else with it? Probably not. You’re probably better off outsourcing the work and just getting the boards cut the way you want when you want another one. But if you’re the type to DIY stuff, and actually enjoy the designing and building of stuff, or if you like tinkering, or want to try a bunch of different layouts and change them often, then probably more so. A CNC machine will allow you to not only make the backdrops (And you can make them out of different woods, not just foam board) but the display cases as well, with carved trim, engraved and carved text detailing information about coins and collections, or even your own “wooden nickels” if you like. Display and storage boxes… If you are the type to do all that? A CNC is overdue in your workshop.
Myself as an example: I have lots of hobbies. Too many, really, but so many of them are things that the CNC makes possible, or at the very least easier and/or cooler. I build stuff. I get a charge out of taking something that only existed inside my own mind yesterday and holding it in my hands today. I’ve done woodworking for a long time, I jumped on the 3D printing bandwagon as soon as it was financially possible for me. I’ve built DIY video projectors, DIY furniture, DIY audio amplifiers, custom car audio, etc. A CNC is an excellent investment to me, because I use it for lots of other things that I enjoy doing, and will continue to use it on a regular basis.