You certainly have a complicated situation. You may indeed need to hire a lawyer - or lawyers - for both court cases. The general rule on venue is that the case should be heard in the county where the child primarily resides. If the answer to that is Queens, the case will likely (though far from guaranteed) be transferred there. If you were arrested, you need to aggresively fight the criminal charges, as if you're found guilty in criminal court, you can't then claim innocense in Family Court. In any event, I encourage you to schedule a follow-up consultation with an experienced NYC Child Custody lawyer.
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Retain an experienced child custody attorney and file a petition for custody and visitation. Follow all court orders. These cases take time. Unfortunately, some people file false allegations to try and gain an advantage. Do not discuss the matter on line, it is not confidential.
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If the matter cannot be resolved by agreement the only thing that can be done is to go to trial. The father should also file a petition for custody. See a family law attorney to discuss the facts of the case.
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This advice is not meant to create an attorney-client relationship and is a general anwer to the question posed.