Is it permissible to maintain ties with a relative who allows his wife and female mahrams to appear before men adorned, to shake hands with them, sit with them, and speak to them directly or over the phone? Is it permissible to speak with that relative about his wife's relationship with some individuals? And does the description of "dayyuth" apply to him?

Maintaining ties with a disobedient relative and advising them is a religious requirement. If the relative allows his wife and daughters to speak with non-mahram (definition) men directly or over the phone, this is a dereliction of his qawamah (guardianship) and wilayah (authority), and a cause for them to fall into forbidden acts. If he persists in this, then he is a dayyouth (cuckold), content with wickedness and corruption among his family.