Can my wife take custody of my son, if she has a criminal record?

Given what you have written about with respect to ypur wife and her past issues she can take custody of your son if you are willing to allow such but another person could have a vald contest of such given what you have written about her and the safety of the minor.

This is not an easy question to give guidance on in this type of forum.  But what I think that you need to do is to speak with an attorney about the types of custody available in Texas (the state you are writing from, correct?). Here is my understanding:

"In Texas, courts divide child custody issues into two different categories: conservatorship and possession and access.Conservatorship is basically the rights and duties of the parents (i.e. to make decisions for the child regarding schooling, medical decisions, and psychiatric decisions, among many other things). Conservatorship can be done in different ways, including allowing one parent to make all the decisions (Sole Managing Conservatorship) or allowing both parents to jointly make the decisions (Joint Managing Conservatorship). When determining the rights and duties of the parent(s), the court will decide what is in the “best interest” of the child, which takes into account a large number of factors. Possession and access refers to when the parents have physical custody of the children or when they can visit with the children. Texas has two statutory possession and access schedules: standard and extended standard. These schedules dictate the time each parent spends with the child. However, the parties can agree on different possession and access schedules based on their needs or the court can order a different possession and access schedule based on the best interest of the child."

You appear to have a good case to ask the court to rule in your favor as the parent who will serve the best interests of the child. Good luck.