That tip provides great information if you stumbled across the authentication issues that I had. Anyway, at the end of it all, I have almost exactly what I wanted (still a few bugs, but at least I can get onto the server now). Below, you see part of my list of folders in the left tree and the e-mails within the folder on the right, below all that you see the headers (which should also display the mail content ultimately).
"But why on earth would you want to create a new mail client?! Don't you already have one?! What a waste of time!" Well, thank you for asking. My ultimate aim, of course, is to port this JavaMail Swing client to the NetBeans Platform. I imagine it will end up being something like Josh's Postfachinspektor.
So, watch this space, because I will explain everything I did/will do and then you can follow along and create your own mail client in NetBeans IDE (or for your own NetBeans Platform application).