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.
A woman’s brother-in-law (sister's husband) is a non-Mahram (permanently unmarriageable) to her, so it is obligatory on her to deal with him on this basis. If she goes beyond the limits of the Shariah, like by shaking hands with him, letting him touch her body, and other matters that are mentioned in the question, then she is disobedient to her Lord and negligent about her husband’s rights.
Hence, she must repent to Allah and refrain from these actions even if her husband does not prohibit her from doing so. If her husband prevents her from doing so, then the prohibition is even more confirmed, as she is Islamically obliged to obey her husband in what is permissible.
Abu Hurayrah, may Allah be pleased with him, narrated that the Prophet said, “If I were to order anyone to prostrate to someone else other than Allah, I would have ordered the wife to prostrate to her husband." [At-Tirmithi]
Also, it is not permissible for her to ask her husband for a divorce without a valid reason, as this is prohibited. Thawbaan, may Allah be pleased with him, narrated that the Prophet said, "The smell of Paradise is forbidden for any woman who asks her husband to divorce her without a valid reason." [Abu Daawood]
Her husband preventing her from communicating with her brother-in-law is not a valid reason for her to seek a divorce. Also, if he prevents her from communicating with her sister for a sound reason, such as fearing that she would corrupt her, then he has the right to prevent her from doing so.
If the husband fulfills the basic obligations towards his wife, such as spending on her and the like, then this is a good reason for her to stay married to him and return this kindness with similar kindness and not become rebellious towards him and ask him for a divorce.
If the marriage is still ongoing, then the children's custody is the right of both parents, but if they separate, then the mother is more entitled to their custody, unless she remarries or there is an impediment that prevents her from fostering them, like if she is dissolute and the like, in which case their custody moves to the female who has more right to custody, such as the mother of the mother [the grandmother of the children from the mother’s side].
If the father fears that she might turn them against him and that she would be negligent in raising them properly, then he should try his best to take them from her.
The scholars of Fiqh stated that the parent who has custody of the children has no right to prevent the other parent from seeing them. The mother has no right to prevent the father from seeing his children, raising them, teaching them, and making arrangements with a teacher to teach them, but they spend the night with their mother.
The Maaliki scholar Khalil ibn Is-haaq said, “The father is entitled to take care of his children, discipline them, and send them to school [or any place to learn].” Al-Hattaab said when commenting on this statement, “This is like the statement of Imam Maalik in Al-Mudawwanah: ‘The father should look after his children when they are with their mother, discipline them, and send them to school; but they should not stay overnight except with their mother.’”
Allah knows best.