Is the mother held responsible for adding a father's name to a birth certificate (who has denied paternity)
Once paternity is proven?

Held responsible in what manner?

Typically when a paternity suit is settled, the birth certificate is reissued by the court to reflect parentage. This is automatic in some places, some places you need to request it be done.

I guess I am not fully understanding what you want to know.

Who is filing against who in court, normally the one who fied will do the petition to the court for a new birth certificate