Where the baby was born is usually not relevant to the issue of child support and custody. In order for New York to have jurisdiction to handle a child support case, there must be or have been some connection between the mother and New York. If for example, the mother never lived, worked or even visited New York and the baby was born outside the United States, New York would have no jurisdiction to originate a support proceeding. If however, the mother lives in a Country that is a participant in various international treaties to help establish child support, she could file the suit in her country requiring you to appear in the New York Courts. New York law is then used to determine how much support would be paid. Spousal support is something completely different since many international treaties concern establishing child support or enforcing child and spousal support.
If there is no prior order of custody and neither the mother nor the baby never ever lived in the United States and the baby is more than six months old, a New York judge would be precluded from entering any order on the issue of custody since New York would have no jurisdiction over the proceeding. The issue is not so much whether the baby was born outside the United States, but rather where the baby resides now and for how long the baby has resided outside the United States.
Advice on this forum is for informational purposes only and should never be mistaken as a substitution for legal advice. Answering a question does not create an attorney client relationship. If you need legal advice, you should consult or retain legal counsel.
Child Support, if based upon the laws of NY or another state, is the same. However, the duration of the child support obligation varies between 18 and 21 years depending on the jurisdiction. If the baby was born outside of the U.S. and is not a U.S. citizen, or has not ever been in N.Y., the custody proceeding has to be brought in the foreign country UNLESS the New York Court that is presiding over the divorce by virtue of having personal jurisdiction over the other parent grants custody to you. You should really consult with an attorney if you are not represented by an attorney now.