Yes, a person who owes child support can still be deported. You don't have to be in the same country to pay child support. You can send checks through the mail.
No but theyll lock you up...
Yes. Many nations have a reciprocal agreement with each other regarding child support payments. If you were deported from the US to the EU for example, your country may still enforce your child support payments towards the US.
All you need to do is, petition the court to modify your child support order.
It can be done if the father is living in a country that has an agreement with the US - contact your State's child support agency.
Contact the child support agency in your area in charge of disbursement/enforcement. They can probably obtain that information.
If he was deported back to a country that has an agreement with the US there is no problem, just contact your State's child support agency. If it's a country the US do not have an agreement with you have to apply in his country.
Either scenario is possible.