You should probably seek the assistance of a Family Law attorney in Nassau County to anticipate a custody or visitation petition. I take it there is no current Family Court order awarding you custody or naming the father in a paternity matter. It is possible that the father could petition for visitation and even shared custody, but it is unlikely he will get custody, based on what you are saying, since that would not be "in the best interests of the child". However, he might be able to get supervised visitation.
Much will depend on the kinds of threats he's made in the past and how recently and how much he's appeared to have "reformed" since his brushes with the law and use of drugs. Visitation, if any is granted, will almost certainly be required to be supervised. He will have to prove paternity as part of these proceedings, but that isn't difficult now because of inexpensive DNA testing.
Still, instead of awaiting papers being served, I'd get out in front of the issue and at least be prepared by consulting an attorney. The strategic issue of whether to apply for custody now rather than waiting to see if and when the father does it is something you should discuss, because the filing of a custody petition will probably require that the father be given notice of the court proceeding and an opportunity to appear.
Hope this helps. Good luck to you!
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All of your goals here can be accomplished. Prepare to defend a Petition for Visitation and/or Custody just to be on the safe side.
You would be well-served to discuss your dilemma with a New York domestic relations attorney in a confidential forum as soon as possible.
If you want to take pre-emptive action, you may seek a protective order because of the threats if they rise to the level of harassment. Coupled with the other negative factors you state about him, it would make it more difficult for him to obtain joint custody and perhaps visitation. You should consult an attorney to discuss you alternatives.
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