Can a woman ask for Khula without giving a reason, and what are the child custody rules?

My brother and his wife were married 12-13 years ago and now my sister-in-law is asking for Khula reasons stating (1) he is not praying, is against Islam, not allowing the child to learn Quran, etc. The first reason is true and now on our persuasion is praying, all other reasons stated by her are wrong. Has she got the right to ask for khula? Some of the intermediates who are talking are saying that a woman has the right to ask for khula without stating the reason as per Quran is it true? Since they have a child who is a daughter of 8 years old incase they divorce who has right to have the daughter the mother or the father. Please briefly tell us the rules of Khula as my brother is not interested in giving away his daughter. On an earlier occasion she has asked for Khula and we managed to convince her, but now she i.e. very adamant. If you can reply on this immediately we would be thankful to Allah.