It doesn't matter that he's not on the birth certificate. However, courts are extremely leery of depriving a child of a parent, even if an absent one, and even one with a criminal record as long as it doesn't involve abuse. Fortunately, it's now become more difficult than in the past to take a child across a border without proven consent of the other parent than it once was, which gives you some protection -- but only if he uses a legal crossing point with authorities there. I gather you're afraid that he won't do that.
Given the large number of factors you have here, you certainly appear to have a case for majority physical custody, if not sole custody. The best thing you can do, however, is to consult a lawyer in your area (preferably your own county) who has handled cases involving an immigrant parent before.
Yes, you can file for primary custody of your child based upon the fact that he does drugs, steals, has been in jail, etc. You already have custody, but filing paperwork (a paternity action if you were not married) will likely trigger a response from him requesting that he have visitation time. Right now, it sounds like he would only be given a monitor. Further, right now, you do not sound like you have any court orders requiring you to give him any visitation time. You could contact the District Attorney and go after him for child support, and then he will likely respond with his own paternity action requesting visitation time. Talk to a family law attorney about what your options are.
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