What if the father is present and does not want to establish paternity? Will they just let him add himself to the birth certificate?

First, its not a good idea to piggyback your question on another thread. To avoid confusion I've moved your post to its own thread.

I'm confused by your question. If the mother and father are not married then they BOTH would have to agree to have him listed as the father. Usually an acknowledgment of paternity would be required.