The birth certificate lists four names, first, middle, father's last name, mother's last name. Can the last (mothers last name) be dropped?

Additional information to consider... the parents were never married. The father has physical custody. The mom pays no support, makes no effort to see or communicate with the child. THe father is now married, he and his wife are raising the child. They both share his last name.. the child has First, middle, dad's, mom's. The child is uncomfortable with the difference.
She is 7 years old.

The child can be called anything they want.

As for official records (school, Social Security, etc.), you would have to go to court to change the name.

If they want to change the child's name, that has to be done in court, but can require the mothers to allow it normally.

Also what about legal custody, has it went though court for him to have custody, does mother pay support