Is it permissible to divorce a pregnant wife who frequently asks to visit her family, and what are the rights and obligations of each spouse in this situation?

Do not rush to divorce your wife, for divorce is the last resort. Your wife's conditions for returning are not binding upon you. You are not obligated to permit her to visit her family twice a month; rather, custom should be observed. You may prevent her from visiting her family if you fear they will corrupt her. If her conduct towards you is poor, begin with admonition, then abandonment in bed, and if that does not work, then a non-severe beating. If these methods do not succeed, then let two arbitrators intervene for reconciliation or separation. If you divorce her, she is entitled to maintenance and housing during her waiting period ('iddah), and her waiting period ends upon giving birth. She is entitled to her full mahr, and a consolatory gift (mut'ah, a sum of money) is recommended. Custody belongs to the mother unless there is a valid impediment, and if a dispute arises regarding maintenance or custody, the Sharia judge is the one to decide it.