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You won't get into trouble for leaving the state with your children. There is no custody order and legal paternity has not been established. You are free to take the child and leave, legally. That said, you can't file anything in Florida until you have lived there for at least 6 months. So, if you were to leave, he could file a paternity action and you would have to come back to NY for the hearings. He "might" be able to get an emergency custody order while the case is pending.
But he can't do anything without filing a paternity action. You can do this yourself as well, in NY. He can't make you stay in NY... but it may hurt you in a custody hearing if you take him to Florida and don't offer any contact or visitation. If you are going to go, just make sure you leave your contact information and a note telling him to contact you if he wants to set up a visitation schedule. That way, if he does file a paternity action, he won't be able to argue that you were denying contact.
The other option is to file it in NY, and then get permission to go back to Florida from the court.
Not necessarily. You can ask for "preliminary orders" which can usually be gotten in a much shorter time frame, generally weeks. But, you are correct, it won't be immediate.
He can't kick you out. You have set up residence, he would have to evict you. Just refuse to leave... or go ahead and leave and go back to Florida. When he files his paternity, you will have a valid defense for taking the child to Florida. As far as what you can do that would be immediate... you could file the paternity and then ask for an emergency custody hearing. It may or may not be granted though, and emergency hearings are not always immediate either. Sorry, I wish I had a better solution for you... but I can't offer you what doesn't exist. An emergency hearing request is the quickest route to a hearing, if it is granted.