So you have current orders, correct? Do the orders stipulate anything about relocation, such as restricting the primary residential parent from moving over a certain distance from a postcode? Did the mother make a reasonable effort to discuss the relocation and educational arrangements with the father before moving? Is the move of such significance that it interferes with the father's ordered time with the child?
I'm unsure about what 'papers' she would have been issued with - an application for a contravention order? A letter from a solicitor?
In any case, she doesn't have to relocate on mere demand of the ex. It's a threat, not an order, but if he pursues proceedings, you will need to show why the move is in the best interests of the child and how you intend on facilitating time between the father and his son if the court were to order in your favour.
Summarily, it's unlikely the father would win sole parental responsibility or residency if the routine has been to live with the mother, but as always, the court will assess what's in the best interests of the child according to Section 60CC of the Family Law Act 1975 and make orders accordingly. In relocation matters, you can also indicate how the move benefits you and by association, benefits the children.
The details you have provided about the father going to parties and the like is fairly irrelevant to children's best interests in the eyes of the court. Drug use may have some weight, but if the child isn't at risk of harm, it's not going to receive a lot of weight. What you want to focus on isn't what a terrible parent the father is (as you have done above), but on how the mother will continue supporting the child's relationship with the father, despite the relocation. Basically, the court isn't concerned with what opinions the mother has about the father, only how the child's relationship with his parents will be facilitated.
Hope this helps!