Rights of Women after Divorce in Islam
Question

I would like to ask about the rights of the wife in the case of divorce when she has custody. Please note that she is the one who asked for divorce.
Praise be to Allah.Asking for divorce due to husband's shortcomings
If a woman has asked for divorce – when her husband has consummated the marriage with her – one of two scenarios must apply: 

Either she has asked for it because of the husband`s shortcomings in fulfilling her rights, or because of some attitude in him that annoys her and deprives her of her rights, or because he has fallen into sin, and other such reasons which make it permissible for her to ask for divorce. The Shar`i judge is the one who should evaluate them and decide whether they are valid reasons or not. If they are valid, then the husband must divorce her by divorce in that case, and give her all her rights in full, which are: 


The full dowry that was agreed upon, both the earlier portion, if any of it is still outstanding, and the delayed portion, because the Messenger of Allah (peace and blessings of Allah be upon him) said: “If he has consummated the marriage with her, then the mahr is hers because of his intimacy with her.” (Narrated by At-Tirmidhi, 1102; classed as authentic by Al-Albani in Irwa` Al-Ghalil, 1840)
Reasonable maintenance, including food, drink, accommodation and clothing during the `Iddah period , if it is revocable divorce.

Shaykh Salih Al-Fawzan (may Allah preserve him) said: 
“The woman who has been revocably divorced is still a wife so long as the `Iddah continues, and she is entitled to the same as other wives of maintenance, clothing and accommodation.” (End quote from Al-Mulakhkhas Al-Fiqhi, 2/317) 
If the divorce is not revocable, such as a third divorce, then she is not entitled to maintenance or accommodation. 
Muslim (1480) narrated that Fatimah bint Qays (may Allah be pleased with her) was divorced by her husband for a third time, and she asked the Messenger of Allah (peace and blessings of Allah be upon him) about what she was entitled to of maintenance. He said:  “You are not entitled to maintenance or accommodation.” 
If she has custody of her children from him, then in that case he must also give her: 

Payment for custody and breastfeeding.
Maintenance for the children.

Allah says (interpretation of the meaning):
“The mothers shall give suck to their children for two whole years, (that is) for those (parents) who desire to complete the term of suckling, but the father of the child shall bear the cost of the mother`s food and clothing on a reasonable basis. No person shall have a burden laid on him greater than he can bear.” [Al-Baqarah 2:233] 
So Allah has made it obligatory for the father of the child to spend on the mother who breastfeeds her child. This includes the woman who is pregnant or divorced; the father must pay for maintenance. (End quote from Tafsir As-Sa`di, p. 105). 
Shaykh Muhammad ibn Salih Al-`Uthaymin (may Allah have mercy on him) said: 
“The woman who is thrice-divorced is not entitled to maintenance from her husband for herself, but he should spend on her for the sake of the pregnancy. On that basis, whatever she needs of maintenance because of the pregnancy, the husband must provide it, and after the delivery, he should pay for the breastfeeding and clothing and the like of the child, but he is not obliged to provide the mother’s food after the delivery. Allah says (interpretation of the meaning):  
“And if they are pregnant, then spend on them till they lay down their burden” [Al-Talaq 65:6].” (End quote from Liqa`at Al-Bab Al-Maftuh, 147/question no. 8) 
It says in Al-Mawsu`ah Al-Fiqhiyyah (17/311): 
“The Shafi`is and Hanbalis are of the view that the woman who has custody is entitled to demand payment for custody, whether the woman with custody is the mother or someone else, because custody is not obligatory for the mother, and if she refuses custody it should not be forced upon her, generally speaking.” (End quote)
Setting the amount of payment and maintenance should be referred to the judge. 
It is not permissible for the husband to put pressure on his wife so that she will give up any of the rights mentioned above. 
Asking for divorce for no reason

The second scenario is where the woman asks for divorce from the husband with no cause. In that case the husband may ask her to return the dowry that he gave her, in return for divorcing her. This is what is called Khul` . 

In this case, the four rights mentioned above no longer remain, except payment for breastfeeding and custody, and maintenance of the child who is still being breastfed. 
`Abd Ar-Razzaq narrated in Al-Musannaf (4/90) that Ash-Sha`bi was asked about the woman who is divorced by Khul` – is she entitled to maintenance? 
He said: How can he spend on her when he takes from her? 
Because if the Khul` is done, the husband cannot take her back, so she is like one who is irrevocably divorced, and is not entitled to maintenance. 
Ibn Al-Qayyim (may Allah have mercy on him) said: 
“The irrevocably divorced woman (unless she is pregnant) is not entitled to maintenance or accommodation according to the authentic Sunnah of the Messenger of Allah (peace and blessings of Allah be upon him); indeed this is in accordance with the Book of Allah and is what is implied by analogy and it is the view of the scholars of Hadith.” (End quote from I`lam Al-Muwaqqi`in, 3/378) 
And Allah knows best.