Role of Mother’s Husband Towards Wife’s ChildrenWith Regard to Their Upbringing and Education
Question

What is the role of the mother’s husband towards the children of his wife? In this case the father of my daughter is not Muslim and he does not want to become Muslim. Can my husband take charge of the issue of educating and raising my daughter? Will Allah ask him how he raised her? How should my husband deal with her when I feel that he does not love her and is not comfortable with her, because she is eight years old and she did not receive any Islamic education in the beginning, so her behaviour is not proper?.
Praise be to Allah.Firstly: 
The small child, whether male or female, should follow whichever parent is Muslim, if the parents belong to different religions. This is the view of the majority of scholars. 
It says in al-Mawsoo‘ah al-Fiqhiyyah (2/310): 
If the parents follow different religions, and one is a Muslim and the other is a kaafir non-Muslim, then their young child, or their older child, if he is insane when he reaches the age of puberty, should be Muslim, following the better religion of his parents. 
This is the view of the Hanafis, Shaafa‘is and Hanbalis. End quote. 
Based on that, this daughter of yours is a Muslim, and the responsibility for taking care of her and raising her rests with you. She has nothing to do with her kaafir non-Muslim father with regard to care and upbringing. 
Secondly:  
With regard to her mother’s husband (your husband), it is unlawful for him to marry this daughter so long as he has consummated the marriage with her mother, because the daughter is a rabeebah (stepdaughter) to him, and the stepdaughter is one of those whom the man is forbidden to marry, so long as he has consummated the marriage with her mother. Allah says (interpretation of the meaning):
“Forbidden to you (for marriage) are: your mothers, … your stepdaughters under your guardianship, born of your wives to whom you have gone in — but there is no sin on you if you have not gone in them (to marry their daughters)”
[al-Nisa’ 4:23]. 
The scholars of the Standing Committee for Issuing Fatwas said: 
It is forbidden for a man to marry the daughters of his wife with whom he has consummated the marriage, and he is regarded as a mahram (relative to whom marriage is forbidden) to all of her daughters both before and after marriage. Allah says (interpretation of the meaning):
“Forbidden to you (for marriage) are: your mothers, … your stepdaughters under your guardianship, born of your wives to whom you have gone in —”
[al-Nisa’ 4:23].
But if he has not consummated the marriage with her, then he is not a mahram for her daughters, because Allah says (interpretation of the meaning): “but there is no sin on you if you have not gone in them (to marry their daughters)” [al-Nisa’ 4:23].
End quote. 
Shaykh ‘Abd al-‘Azeez ibn Baaz, Shaykh ‘Abd al-Razzaaq ‘Afeefi, Shaykh ‘Abd-Allah ibn Ghadyaan, Shaykh ‘Abd-Allah ibn Qa‘ood 
Fataawa al-Lajnah al-Daa’imah, 17/346, 347) 
So your husband is a mahram to her, but the responsibility for the upbringing and care of this daughter does not rest on his shoulders, unless he does that willingly and as part of treating her mother kindly. And this is what we advise him to do, so that he might earn a great reward, especially if he knows how great the need is for such a teacher and educator, and that her father is not qualified to undertake this mission, and that her mother is too weak to undertake this responsibility on her own. He has a good example in the Messenger of Allah (blessings and peace of Allah be upon him), as he married women who had children and he took charge of their upbringing and caring for them. There follow some examples of that: 
It was narrated that ‘Umar ibn Abi Salamah said: I was a young boy under the care of the Messenger of Allaah (blessings and peace of Allah be upon him), and my hand used to wander all over the plate. The Messenger of Allah (blessings and peace of Allah be upon him) said to me: “O young boy, say the name of Allaah and eat with your right hand, and eat from what is nearest to you.” 
Narrated by al-Bukhaari (5061) and Muslim (2022) 
‘Umar ibn Abi Salamah was the stepson of the Prophet (blessings and peace of Allah be upon him); he was the son of the Mother of the Believers Umm Salamah (may Allah be pleased with her) from Abu Salamah; he was born in Abyssinia when his parents migrated there. 
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on him) said: 
Among the lessons we learn from this hadeeth: it is obligatory for a person to discipline his children and teach them how to eat and drink, and to teach them what they should say when eating and drinking, as the Prophet (blessings and peace of Allah be upon him) did with his stepson. In this hadeeth we see the good attitude of the Prophet (blessings and peace of Allah be upon him) and how he taught others; he did not rebuke this boy when his hand started wandering all over the plate; rather he taught him gently and spoke to him gently: “O young boy, say the name of Allaah and eat with your right hand.” End quote. 
Sharh Riyaadh al-Saaliheen, 3/172 
Your husband should understand that his wife’s daughter is still very young, and that he can succeed in bringing her up to be obedient and chaste by winning her over with kind treatment and gifts, and speaking kindly to her. If friendship develops between the two parties and love grows strong between them, he can influence her in ways that will make her attitude and behaviour better. You also have an important role to play in making her dear to your husband and strengthening the relationship with them, helping your husband to be patient in dealing with her, and reminding him of the great reward that there is in bringing her up and taking care of her. 
We ask Allah to help you to do that which is best for you, to make you steadfast in following true guidance, and to guide you to the way of modesty and piety. 
And Allah knows best.