You can get wifi from Etisalat or Du (some areas in Ajman don't have wifi, so check beforehand). The electricity and water is from "FEWA," and the rates are similiar to Dubai. Ajman City Centre is a pretty good mal (Carrefour, free wifi, cinema, some shops, decent food court, etc.)
You need your divorce paper, the children's original birth certificates and passports, the father's valid passport copy, a document stating you are the legal custodian of the kids. Stay strong.