The said child is that of the father by whose sperm he was conceived, as well as being the mother's child. The child belongs to both of them, but does not inherit the father as he was born illegitimately.

There is no doubt that there are two types of conjugal relationships between males and females. The first, which is common among societies, is based on marriage, and the children from the couple are legitimate. The second is based on mutual interest and without marital bonds, and the children from such relationships are not considered as legitimate from the Islamic perspective.

Marriage is a common concept between humans and not something that is hidden. If the couple intended to form a relationship of marriage, then they are married even if they did not have a verbal contract between them, and the children then are considered as legitimate. If the couple were intending to have a physical relationship, i.e., sexual intercourse and emotional closeness, the relationship is not considered as marital relation, and the children are considered as out of wedlock.

A number of things are reported to be recommended following the birth of a child, amongst them are the following:
1. Bathing the baby as ghusl for him. Apparently, the one bathing the child is required to have the intention of ghusl.
2. Covering him with a white cloth, and it is disliked to use a yellow one.
3. Reciting the Adhan of the prayer in the right ear and the Iqamah in the left ear.
4. Placing water from the Euphrates River and a small amount of the earth of the grave of Imam Hussein (peace be upon him) on the top and the bottom of his mouth. If the said is not available, rain water can be used instead. Also it is recommended to place honey and date on his jaw from the outside.
5. Choosing a good name for him. It was reported that the most truthful names are those which indicate servitude to Almighty Allah, like Abdullah and Abdul-Raheem. And the best names are the names of the prophets (peace be upon them) and their best is Muhammad. It has been reported also to name the baby "Ali". It is also reported that poverty does not enter a house where its residents are named Muhammad, Ahmad, Ali, Hasan, Husein, Ja'far, Talib or Abdullah, and amongst the females Fatima. It is also reported that it is recommended to name the baby boy at his birth "Muhammad" for at least seven days, and after that either continue naming him that or change it at that time.
6. Cutting the hair of the baby on the seventh day, donating its weight of gold or silver to the needy.
7. Circumcising the boy on the seventh day.
8. The "Aqeeqah": the sacrificing of a sheep, a goat, a cow or a camel. It is better to choose a male animal for the boy and a female animal for the girl.

Some narrations state that it is recommended to name children after the names of the prophets and the names that refer to the servitude of the Lord, like "Abdullah". Also, we find emphasis in some reports on certain names like Muhammad, Ali, Hasan, Hussein, Ja'far, Talib and Hamza, and amongst the female names, Fatima.

It is recommended to circumcise the baby boy at the age of seven days, and it is disliked to delay it. When the uncircumcised male reaches the age of adulthood, he must get himself circumcised. Circumcision is mandatory for Pilgrimage (Hajj).

Yes, you can.

There is no recommendation for the man to shave his head when his Aqeeqah is slaughtered.

The right of the custody of the children is for the mother for the first two years of the child's age. After that, it is obligatory on her to give the custody to the father if he asks for it. At the same time, the father should not prevent the mother from having a relationship with her child, as this is a right for both the child and the mother.

It is not permissible for the father to prevent the divorced mother from visiting her child and see him, except when it is in the best interest of the child to do so.

If the father of the children became an apostate, then his guardianship over the children is no longer valid.

Yes, if she reached the age of nine years, she can decide with whom she wants to live.

If the girl reached the age of Sharia-defined adulthood, she can choose to live with her mother or father. Neither of them is allowed to prevent the girl from seeing the other parent.

Providing for and raising a non-Mahram child is permissible; in fact, it is highly recommended if the child requires such care. Rather, it may even become obligatory if the child’s life depends on it. However, rulings related to the father-son relationship, such as inheritance, will not apply.

It is permissible to adopt a child, and with adoption we mean providing for and raising a non-Mahram child. However, he or she remains a stranger to the family, so wearing hijab is obligatory as you mentioned in the question.

Yes, they can. It is not disliked for them to do so.

It is permissible for others to raise the child in special circumstances like if the parents are deceased or are non-Muslim.

It is not permissible for the father to beat his child, except for the sake of discipline and correction. This must be limited to the need to achieve this purpose. It must be applied gradually with reasoning and should not exceed the need for the sake of revenge and retaliation or due to belittling the child because of his weakness. One must be careful to discern between the welfare of the child and his own emotions such as anger or frustration. The narrations of the Imams (peace be upon them) report that it is permissible to beat the child five or six hits, while emphasizing on leniency. One should not exceed this number unnecessarily.
It is not permissible for the father to beat his son or daughter after they reached the religiously recognized age of adulthood. He must limit his means of discipline by advice and suggestion.

Do not give up, and do not lost any hope in guiding him. Try to correct him by, maybe, getting more close to him and giving him more attention, so he might trust in you more and have more confidence in you and in your fatherhood, or by making friends with families who are religious, so they can also be a positive influence on him. If you continue in this manner then in time he will respond to you positively.
It is natural in for a child in this age to pass through a phase of rebellion, especially in these times and circumstances in which there is a lot of emphasis on personal freedom.