I solved the issue - thanks!
Just for the protocol:
I checked the settings you mentioned already before and compared them to the provided screenshots - they were correct.
DAVDroid was not affected by Battery saving and permission to access calendar and contacts were granted. (There is one screen that looks different on my phone though and that is the “additional permissions” section. At your screenshots you enable permissions for “read tasks” and “write tasks”. Unfortunately, I don´t have this menu items in my list of “additional permissions”.)
I had to restart my phone and once manually trigger the synchronisation inside DAVDroid, then the events suddenly appear inside my calender app, even without adding the calendar account in my calendar app. I suppose that the problem was that I have set the permissions “to late” where DAVDroid had already tried to do a sync which presumably failed. Maybe thats why the restart and manual sync solved the issue.
Thanks a lot!