If my son is 12 and his mother made him move out of state but is not living with her and she is still getting my payments, can she do that by law?

Yes, until you go to court and ask for custody of your son or the person who has him now asks for child support from you.

Yes, but you can move/petition to cancel payments of child support to your wife if the child is not residing with her and she's not paying the child support you pay her to the person(s) who is/are providing care and supervision of the child.