However, I had to switch back to CalDAV-Sync and CardDAV-Sync because DAVdroid is not working for me 
DER GEHT!!!
It works!!! After another hint from the posteo support I checked my router settings. There I have set up two accesses, a main access and one guest access. So far I was using the guest access to access the net. Checking the settings there showed a possibility to restrict I-Net access only to surfing and mailing. After unchecking the respective check box I tried to use the previously set up DAVdroid account to synch my data and … IT WORKS!!!
So now I am a happy DAVdroid user and posteo customer. What else can I ask for?
Thank you so much for your time and patience in helping me analysing this problem although it was on my side and not on yours. Sorry for any inconvenience.
Just wanted to let you know.
Thanks again. Topic can stay closed.