I was able to solve that problem.
MPD port was double busy. Apparently 2 instances of Mopidy were running.
Now the module MMM-MPD runs under a few specials:
During the installation the folder name has to be changed according to the config file.
The config file value maxRows must not be 0!
The display will only update after a song change!
It was hard to find this specifics, but I thank everyone for the help!