in theory he could but as a practical matter, it would be hard as the felony might very well disqualify him from living in the USA. you probably need to at least consult an immigration lawyer in your area or at least dallas. elizabeth carillo is a good one in dallas.
The advice that I give in each answer or legal is not intended to take the place of an in person consultation. A complete answer takes an in depth interview. After all, it is a life that is at stake. If you are in another city that I do not service ask me and I might be able to recommend you an attorney there. In general, in Houston, I recommend Adan Vega or Bruce Coane, Specialists. In Dallas I recommend Richard Fernandez or Yong Wood highly skilled and experienced.
This depends on the circumstances of your case. Custody cases are typically evaluated based on "the best interests of the child" so, in theory, it is possible that the judge could find that the child is better off with you than with his mother, despite your felony. Is it likely? Probably not. You need to consult with a family law attorney to determine what your options are as far as getting custody.
Being illegal will not necessarily bar you from having custody of your child. Many of my immigration clients who are not in legal status have gotten custody of their children. However, as my colleague has pointed out, your felony may make you deportable, so you would also be well advised to speak to an immigration attorney who has experience with criminal matters.