Can the judge grant the mother permanent custody of her 5-month-old child in the event of divorce, and what rights and duties does the mother have, and what about child support, especially since the father has not been spending on him since birth?

If the marital relationship is ongoing, custody is a right shared by both parents. If a divorce occurs, the mother has a stronger claim to custody as long as she does not remarry. The child's maintenance is obligatory upon the father, and he is not permitted to be negligent in this regard. If the grandfather spends on the child with the intention of claiming reimbursement from the father, he is permitted to do so. The wife may claim her maintenance from the husband unless she is defiant (nashiz). A woman is not permitted to seek a divorce except for a legitimate Shar'i reason, and if reconciliation is possible, it is preferable.