Thank you for your detailed answer.
Unfortunately I already did ALL of this except of using the luci-reload command in CLI, there I used just "reboot"
I have to add that I own a Bachelor Degree in Computer Science so I know how to operate a CLI. I was not able to set up the router within two days though....
So step by step. I updated the firmware right out of the box. It does not make a difference whether I use auto APN or internet.telekom. I thought maybe the DHCP/DNS setting of the clients was the problem but the router himself can not resolve the hostnames for the built in speed test as well. I have had an external LTE antenna attached to the router. The weird thing I noticed is when I switched from LTE only Band to Auto selection, the modem was only searching for 3g and was stuck in a loop then. Only resetting the device resolved the issue. And on top, the LTE modem stopped completely from time to time.
I tried several times to reset and started all over from the factory settings but this made no difference. Obviously the easiest way to set up DNS is using the ones by the provider. But even if not I got stuck in setting up DynDNS (because of a shared public IP) or setting up 8.8.8.8 for example because I can not ping outside the network.