Not a problem anymore. I got help in a chat with a Microsoft agent, and after a long session of not knowing what each other was talking about, something happened and I got what I wanted, namely the ability to go to our six email accounts at the server!

The only remaining mystery is that I can access these accounts by going through LastPass, but for the time being only via Internet Explorer, but not via Firefox. Since I have something that works, I'm in no hurry to solve this mystery, but I am curious and I'll pursue it in due time. It's probably an add-on missing in Firefox or something like that.

Unless someone has an idea about the mystery, this thread is closed.