So basically I had no ticket for the first month, don't have one as of now. From my understanding, if I'd take the train and gotten checked it would be ok in July (they stated in the delay email that the email is proof of contract and the people doing checks are instructed accordingly and to accept it until end of July). However, I've already read stories that people still had to pay the 7 Euro administrative fee for not bringing a ticket (but having one in theory, so not the full 60 euro penalty).
For my upcoming train trips in August I will simply buy a ticket and ask for a refund once my ticket is finally valid. The only other option would be to take the train and risk a nice argument about having a valid contract but no ticket.
Simply ridiculous. I could go to a train station with an office and see if I can get it sorted there but I'd need to take the train to get there (only a small station where I live). Also a stupid waste of time.
So far the most amazing business process I have experienced in my life :P Should have just bought the ticket elsewhere (locally, deutschlandticket.de etc.) but no stupid me wanted to have everything in one app and didn't really expect the official app to cause any problems.