Okay now.
I repaired some of the broken methods for testing. Simple Methods such as findCustomerByCustomerId() via GET which gives back a customer object by sending the id. And createNewCustomer(firstName, lastName, city) which creates a customer by POST with these parameters and gives back the created customer id.
In the plain Java class, calling it from my local Java app, both methods worked now.
Calling these methods in Web Dynpro (by an instance of this plain java class) the first method still sends back a 403 error.
And the second one returns a 411 Lentgh Required error. I added service.header("Content-Length", "0") but still I get the 411.