Thank you for your answer - that wasn't the problem. The directory wordpress was installed in wasn't the root directory of the server (although it seemed so) . I had to move it up 2 directories. By the time you looked everything was in change.
AND domain mapping isn't working with UMLAUT Domains - the ä in the domain name was the problem. Now it's redirected to http://allgaeuhoch5.de - and that is mapped and working fine!
But again - thanks for looking & trying!