So it basically compiles a new lwip static library and uses that for linking (instead of talking the one from the SDK).
Yes, once I have the correct DNS server published and open a server on UDP port 53 DNS requests are coming in. The problem now is answering these DNS requests properly. Unfortunately I did not find a quick and easy explanation or example yet, how such a DNS response would look like. So I can not verify if that solution works properly. I hope I will find some time on the weekend to read through the DNS specification.