You have lots to think about
I really want to address your questions and make some comments you need to think about.
1. Does he have rights? Absolutely he has rights and will be able to enforce those rights if he seeks to establish paternity. If he seeks to establish paternity you will have to do a DNA test. If he is the father, there WILL BE A SUPPORT ORDER, and he will have visitation rights. More importantly than that, your child has a right to know his father and you are being shortsighted in trying to deny him that. Your child also deserves the financial support that his father can give him up to and including college. Your child also deserves to share in this man's inheritance.
2. Your boyfriend can not adopt your son.
3. If you MARRY your boyfriend, there are different rules in each state about when he can adopt as the step parent. However, before he can adopt, you have to serve notice on the punitive father and he must give up his rights to his child or you can have those rights terminated for cause (like abandonment). If you attempt to have your husband adopt the child without notifying the punitive father you will be in contempt and can go to jail.
4. If you ever receive Medicaid, or Welfare the state will probably seek to recover from him in the form of child support. That will bring him into the picture.
The biggest thing you should consider is that you are robbing your child of his father. I don't buy that you are doing this, "for his own good." He needs a father, he needs to know the truth, and his father needs to share in the support of this child. It's pretty selfish to not take the child's need to a father into account. Boyfriends don't count. Until he is willing to commit to you, he sure isn't going to commit to your child.
By the way, the child's father could always come after you for CUSTODY too. So you might want to consider just getting this taken care of. Get a custody order while you are in good shape, get a child support order, and let this child see his daddy before you end up losing the child to this married man.
I am not trying to be judgmental, I have had affairs of my own, and I am the adopted son FROM an affair. You have already made a bad decision to have an affair with a married man that resulted in a child. Please don't compound that mistake by alienating your son from his father. Your son will not appreciate you doing that when he is old enough to understand.
Good luck.