Yes, but I teach fathers how to change that.
Guardianship, not custody
A father can get custody of the children if it can be demonstrated that that is in the best interest of the child. The father being the primary caregiver would help.
If she has custody of the kids. Legal custody.
yes
You have the visitation rights that were established in the divorce, and you have no custody rights.
If a child's mother has sole custody then the courts have decided this and there is a reason for it. Generally, if the mother is a good mother (fit mother) then the children will be looked after by her with the father having partial custody to see his children. If the father is unfit, then the mother would have full custody. If you are the child asking the question and are upset because your mother has house rules and you don't always agree with them then this is not a good reason to want to live with your father. Your mother gives you these responsibilities so you will learn good characteristics that will make your life a whole lot simpler. You may not see that now, but will in the future. If this is the father asking the question and you have partial custody the courts have deemed it this way for a reason. If the mother is unfit and the father wants to take full custody then you should retain a lawyer asking for full custody.
Does she want custody or access?
There is no law that stops the mother from going away with another man and leaving the children with the father. However, these circumstances would make it difficult for the mother to win custody of the children in court.
If the mother has interrupted her career more than the father for her children, surely she has the better argument for custody? Custody should depend on how the children are treated, not on how hard the mother found it/finds it to get a job. The father would probably have to work less when they have custody of the children anyway.
Generally, they would go take the father to court and claim for custody (protection/safeguarding) over the children.
The father will find it easier to get the custody of the children , as the wife has left both the husband and as well as the children and is least interested in them.
The mother, unless proven to be unfit or incpable of caring for the children, and regardless of her marital status, invariably gets custody after a divorce.