Being Muslim, for custody cannot be given to a kaafir.

Being of sound mind, for custody cannot be given to one who is insane or mentally deficient.

Being an adult, for custody cannot be given to a minor.

Being able to bring them up properly, for custody cannot be given to one who will be negligent.
Ibn al-Qayyim said:

No custody can be given to a kaafir over a Muslim for two reasons:

1 – The one who is given custody of a child will be keen to bring him up in his religion and to teach him about it, and after he grows up it will be difficult for him to change. He may change him from the natural disposition of man with which Allaah has created His slaves, and after that he may never come back to it, as the Prophet (peace and blessings of Allaah be upon him) said: "Every child is born in a state of fitrah (natural disposition of man), then his parents make him into a Jew or a Christian or a Magian." There is no guarantee that the one who has custody will not turn a Muslim child into a Jew or Christian.

If it is said that the hadeeth only applies to the parents, we should say that the hadeeth refers to the usual situation, which is that a child usually grows up with his parents, but if one or both are lost, then the child's guardian among his relatives takes their place.

2 – Allaah has severed the ties of closeness between Muslims and kaafirs, and has made the Muslims the close friends of one another, and the kaafirs the close friends of one another. Custody is one of the strongest reasons for creating close bonds, which Allaah has severed between the two groups.
Zaad al-Ma'aad, 5/459.
And Allaah knows best.

He Married Her After She Became Muslim But She Does Not Pray

Islamic Rulings - Living Shariah Verdicts

Islamic Questions & Answers

I am a rational Muslim. I told a Christian girl that I would never be able to marry her because of the difference of religion between us. She agreed to enter Islam and I married her believing that she was a Muslim. We have now been married for three years, but it seems that she does not want to pray despite all my efforts. I bought her some Islamic books on this matter. During Ramadan, she started to fast with difficulty but she is not praying up till now. What should I do (if she carries on as she is)? I have started to feel tired of reminding her to pray.

Praise be to Allaah.

In this case you have to leave her, because in this situation she has entered Islam and then left it, and this is regarded as kufr, rejecting the sound religion and apostasy from Islam. If she insists on refusing to pray and fast and do everything that is required of the Muslim, and she has not given up haraam things, then we think that you should leave her. Whoever fears Allah, He will grant him a way out. Allah says (interpretation of the meaning): "But if they separate (by divorce), Allaah will provide abundance for everyone of them from His Bounty" [al-Nisa' 4:130].

Shaykh ‘Abd-Allah ibn Jibreen (may Allah have mercy on him)