Is a father legally allowed to take his child if he's not on the birth certificate. And the father is paying childsupport

Legally< really need to define that, can he legally have his child for visitation as long as the mother approves it , yes.

Legally, being on the birth certificate does not give any specific visit right, what he needs is a court order for child visitation, He can visit with the child against the mothers wishes with a child visit order.

Is the child support though the court ?

Visitation and child support are not connected, one has nothing to do with the other

If he is paying child support by a court order, then he was likely declared the legal father. If there is no other court orders for custody and visitation, then he has equal rights.

If there are no court orders, then he can go to court, establish paternity and get custody and visitation.

ANY question on law needs to include your general locale as laws vary by area.