Parental rights and termination on the basis of abandonment or in the best interests of the child. . Divorce and parental rights are separate issues in reality. Your observations of the INS issues are one reason why domestic jurisdiction is preferable, in my opinion. The child's status is contingent upon the father's.
Dad has just flown the coop, and now his residency status is very questionable starting at six months. Your interest in being his new adopted parent or guardian with the permanent custody is commendable. As a ward of the court, the child can be adjusted in status if you are merely court-appointed guardian. The US gov't will have become the child's protector against any foreign aggressors and only it will have the authority to "send" him back. Under international law, he is an "abandoned foundling" under 18 which officially means he is not deportable. Daddy can most rightfully attempt to reclaim him in US Court, but this is not very likely based on your characterization of his actions. Becareful here, as the INS would love to prove "marriage fraud" on top of your shared fiscal fraud. I seriously doubt that past intent of marriage fraud is the case by his abandonment of his green card status, but they will do anything to become retrospective.
Your situation sounds like four monkeys, not one. That is fiscal issues, marital property, parental rights, and then divorce. The immigration matters are woven into these. I'd get the fiscal debt off your back with either an offer-in-compromise (OIC) or a Chapter 7 bankruptcy. That is a lawyer makes an offer which stops the collections process until they accept or reject. Bankruptcy will also stop the collectors, too. This requires a tax debt over three years old, I think. If you own a NV house with some equity under $125,000, your fiscal matters will not be excluded by state homestead exemption. However, fiscal debt is shared by state community property law (and/or joint return)! This might require a Chapter 13 bankruptcy, so you can keep a roof over this kid's head. A more hybred bankruptcy is Chapter 20 which is Chapter 7/13 together. That "Chap. 7 liquidation" of credit cards and any qualified fiscal debts, then "property liens" are reinstated so that the Chapter 13 "repayment plan" can be established. This going to disappear sometime, so use now, not later. Again, a local lawyer is needed for about USD$600-1500. Either OIC or bankruptcy will get Uncle Sam off your back, and any continuing harassment is punishable by contempt of court.
Concurrently, divorce requires no cohabitation for over one year in Nevada. Parental rights are the "abandoment" issue. I don't know how they handle these in each state but permanent custody as court-appointed guardian is one issue of the step-parent relation on the grounds of paternal neglect. By this definition of child neglect, there is no child support, contact, or visitation by the father during prolonged periods and even an indifferent attitude to you and his offspring. Physical abuse is another form of child neglect but such an aggravating allegation must be truthful. However, that machisismo propensity for domestic violence is a factor for your safety and the childs. Mandatory obligations of the state court is to keep the peace.
No one can be forced to remain married, and while he might be "served" with divorce papers, he might not reply. If he does, he will be potentially facing "back payments" for fiscal and child support. Or he might not be found for purposes of legal service, and a default judgement is ordered. In either case, he might be able to challenge the order in the future. Meanwhile, your own community property issues will have been discharged or resolved. His aren't. Those outstanding government debts of fiscal and child support are called "shark repellant" and could keep him out of the country for over 12 months. Under INS rules, he just got disqualifed for permanent residency by operation of US immigration law and the grounds for that actually started at six months. The deck is not stacked in favor of "Pete Rose". Lord only knows just what community property issues might be left over from a divorce decree, after your OIC/bankruptcy, for DR property attachments for months of uncollected child support.
Until you are in the clear, forget about chasing him. Deal with the four monkeys before they become the four horseman.