All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad is His slave and Messenger.
We will answer your questions in the following points:
The first point:
Females are more deserving and more entitled to have custody of the children; it is for this reason that the mother has more right to the custody of her child. If she marries or there is any impediment that prevents her from keeping custody of the child, then the custody is transferred to the next female who is worthy of it according to the sequence order mentioned by jurists (scholars of Fiqh). The father may deserve the custody of the children with certain conditions mentioned by the scholars, such as, for instance, him having a female who is fit to care for children; for more benefit, please refer to fatwa 84618.
The second point:
Both parents have the right to see the children in custody, and neither of the parents has the right to prevent the other from seeing the children or visiting them, but they do not visit whenever they want or however they want. There are criteria for visitation. Some jurists stated that this depends on the custom in their country. For more benefit, please refer to fatwa 84575.
The jurists of the Hanafi School stated that the parent has the right to see their child who is under the custody of the other parent every day. The footnote of Ibn ‘Aabideen reads, “If the child is with her (the mother), then she should take him out to a place where the father can see him every day.”
As regards taking the child away for a long time, this is not permissible except with the permission of the mother. The fact that the father lives abroad does not justify him taking the children for a long time without her permission. It is better that both parents reach an understanding on this. In case of conflict, then it is for the judge to regulate the matter of seeing the children and visiting them.
The fatwa of Al-Azhar reads, “If it is not possible to organize the matter of visitation according to the agreement of both parents, then the judge shall arrange this provided that it is in a place where the child is not harmed psychologically.”
The third point: If the custody is with the mother and the father wished to travel with the children just for a visit and not to settle there, then this does not waive the mother's right to their custody, so the right of custody remains hers, and it is not permissible for the father to travel with them without her permission. Ad-Dardeer from the Maaliki School said in his commentary on Mukhtasar Khaleel, “If traveling was to settle elsewhere and not for trade or visiting and the like, then he does not have the right to take the child with him; rather, the mother is entitled to the custody and keeps the child with her, and the guardian leaves the child with the mother.”
The fourth point: The father has the right to take care of the children and to educate and to nurture them regardless of whether the parents are still married or are separated and the custody was with the mother. Al-Hattaab, from the Maliki School of jurisprudence, said in Mawaahib Al-Jaleel, “This is like the saying of the author in Al-Mudawwanah, ‘The father has the right to visit his child who is with his mother, discipline him, and send him to school, but the child stays with his other only overnight.’”
Therefore, the father has the right to determine the school that the child will join, and it is the interest of the child that is taken into consideration in this regard.
The fifth point:In the statements of the scholars, we have not come across any mention about the right of the father to be informed about everything related to the matters of the child in custody regarding his upbringing, education, health, and so on. In principle, the mother is entrusted with the child, knowing that his education and his health is the responsibility of the father.
Finally, there should be an agreement and understanding between the parents regarding the affairs of the children and what is best for them. If conflict happened to take place, then it is the Islamic court that settles it.
Allah knows best.