What are the rights of the stepfather in regard to discipline, etc.

Please tell me the rights of the step parent of a child.

Whatever the mother and the stepfather agree on.
Whatever you do you do not want to undermine the mother.

The mother can sign for the step father at school and the doctors office giving the stepdad permission to get the child from school or whatever they agree on that is acceptable to the school and doctors.

Yes, you may do anything the mother would agree you can do. Normally I would say you should not spank the child but let the mother do that. But there is no reason you can not ground them, send them to their room, and the such