Go to a lawer.
You cannot take a child out of the country without the consent of the other parent.
Not if your the parent taking care of the child.
Yes this applies to all kinds of traveling out of the country with a child.
Under Australian the non main carer can take a child out of the Country on Holidays. The child must have a passport. To obtain a passport, both parents need to sign the application form. If the other parent will not consent to the application for a passport then an application to the court must be made. If the child has a passport, then the holiday can be taken provided that it is in accordance with the Orders of the court, or by agreement with the other parent. If you are worried about the other parent taking the child out of the country without your agreement, then you can apply to the court for the child to be put on the airport watchlist.
You can prevent it. Will it work? Most likely not.
The parent would be charged with kidnapping.
What should a notarized permission letter to take a child out of the country with one parent say?
As in taking the child away from the other parent? Perhaps
Yes. A parent cannot remove a minor child from U.S. jurisdiction unless he or she holds sole legal and physical custody of the child. A passport will not be issued for the child unless aforementioned conditions exist and/or the non custodial parent personally appears at the hearing and signs an affidavit agreeing to the action.
If the child is a minor, yes.
If you can legitimately claim the child, do so. Let the IRS worry about whether the NCP is claiming the child.
A noncustodial parent can prevent the custodial parent from leaving the state with a baby or child. The court will decide if the custodial parent has just cause to leave the state.