Is it permissible for this divorced woman to marry عرفيًا (عرفيًا: customarily/unregistered, but according to Islamic rites) —to maintain her chastity and due to fear for her children from her drug-addict ex-husband— given that the groom's family is aware of this marriage, and her guardian refuses her marriage?

Many parts of the question require referral to the Sharia court. An 'urfi marriage (customary marriage) is valid if it fulfills the conditions of validity, such as the guardian (wali) and witnesses. It is not permissible to conceal a marriage in order to retain the mother's custody. If the mother remarries, her right to custody is forfeited and transfers to the next person in line according to priority. If the father refuses to arrange the marriage [of his daughter], the matter should be referred to the judiciary. The father is not entitled to custody if he is an addict or has a mistress, as فسق (moral corruption/wickedness) forfeits custody. Some jurists stipulated for the father's custody to be obligatory that he must have a suitable female [relative] for custody.