You (and google and everybody else on the Internet 😉 ) convinced me it was a server-related problem
so I contacted the hosting company again. This afternoon we did something drastically and I asked them to change the server. I was on a windows-server and now I am on a linux-server.
Behold… the server-error is gone and I can finally start building my site!
Thank you for your patience and trouble! 🙂