After receiving a subpoena to have my child custody disputed and for me to pay child support how do I make a rebuttal to the allegations that were presented?

You either address the issues or at least to me, I would hire an attorney to first see about getting the motion dismissed if it has no merit, or to represent my rights in court.

Does the other party have an attorney?

If the other party has an attorney then you would be well advised to get your own attorney.

But you respond with whatever proof you have to disprove the allegations.

After receiving a subpoena to have my child custody disputed and for me to pay child support how do I make a rebuttal to the allegations that were presented?

The caption to this thread mentions a summons, but you say in your post that you were served a subpoena. Those are two different things.

A summons begins a lawsuit and is normally accompanied by a complaint. You respond to a complaint with a pleading called an answer, which admits those paragraphs of the complaint that are true, and denies the others. You sign this and file it in court, sending a copy to the plaintiff, or his attorney if the plaintiff is represented by an attorney.

A subpoena, on the other hand, requires you to present yourself at a certain place and time in order to be examined under oath (give testimony).

Which is it?