leverage the network internal dns service (which does not exist for the default network) and use the service or container name of the container to address it.
Does it matter if I make it public ownership in the same way as the default network or leave it as Admin ownership?Every custom network has its own dns service by default (i am not even sure if it's possible to disable it...).
Just the default network (docker0) does not happen to have one. This is why container in the default network need to be "linked" to access another container in the same network. The container link will create an entry in /etc/hosts for the linked container. This is an annoyance from the early days of docker.