Questions regarding divorce in return for letting the husband off the mahr or some other obligation, divorce during menses, and denying paternity of the child
Question

There is a husband who does not fulfil his duties and responsibilities towards his wife; rather instead of doing so, he mistreats her verbally, mentally and physically, and instead of spending on her he takes her money and spends it on his own interests. After all this suffering she is compelled to ask him for a divorce, but he refuses, then when her family intervenes, he says that he will divorce her, but that is on condition that she lets him off paying the mahr, and her family agreed to that and wrote a document on her behalf in which they agreed that she will give up her right to the mahr if he divorces her. At the same time, he wrote a document in which he said, “In light of the request of wise people among the family, and because of the misunderstanding that occurred between me and her (i.e., his wife), I am divorcing her thrice, and there will never be any relationship between me and her in the future, and I do not have any child from her.” This is despite the fact that he had slept with her two days before that, and he knew that she was pregnant from him. Therefore the questions are:
Is this talaaq or khula‘?
If it is khula‘, does it come under the same rulings as talaaq with regard to taking the wife back? In other words, is there any chance of taking her back?
If it is a talaaq, is it irrevocable and is it the case that he cannot take her back except after she has been married to another man? Or is this threefold talaaq only counted as a single talaaq, and therefore does he have the right to take her back?
If it is a talaaq, then what about the mahr that her family forced her to give up to him? Does he have to pay it despite what was written about giving it up? As far as we know, the husband is excused from paying the mahr if the wife is the one who is asking for separation for no good reason, but he forced her to do that with his bad behavior and bad treatment, so I think the ruling is different. Of course this applies if we regard this as a case of khula‘, but if it is a talaaq then it is more appropriate that he should pay, isn’t it? From what we know also, talaaq must take place during a period of complete purity (from menses) in which he has not had intercourse with her, and that he must wait for one month after that, and the same applies in every talaaq until the three talaaqs have been used up, then after that he should pay the mahr; but if he says the three talaaqs in one go, that only counts as one talaaq. But there are some scholars who say that it counts as a threefold talaaq, hence she is irrevocably divorced. What is the correct opinion on all these details? I hope that you can clarify, because we are people with little knowledge, and we cannot find anyone we can rely on with regard to these sensitive matters except you. As far as I know, Allah hates divorce, hence He has allowed man three opportunities in which to rethink the matter and take his wife back. Perhaps anger may overtake him in the first instance, in which case he can take her back. But some of the Sunni scholars here say that divorce is divorce, and that it takes place in all situations; so long as he has uttered it three times, then it counts as a threefold divorce, and they quote some hadeeths as evidence for that. We hope that you can advise us about this matter. Finally, in that document the husband wrote that he had no child from her, at the time when she was carrying in her womb a child from him, and she will give birth to this baby seven or eight months from now. What are the rulings connected to that?
Praise be to Allah.Firstly: 
Undoubtedly this husband – if he really did that – is mistreating his wife, depriving her of her rights, and going against what Allah, may He be exalted, has enjoined upon him of treating her in a good manner. He should have been kind to his wife and given her her rights in full, and if he disliked her, he should have parted with her in a good manner, in obedience to the words of Allah (interpretation of the meaning): “either take them back in a good manner or part with them in a good manner” [at-Talaaq 65:2]. This husband is not keeping her in a good manner, because he is mistreating his wife, transgressing against her and wasting her money. And he is not parting with her in a good manner, because what he should have done, if he wanted to separate from her, is to give his wife her rights in full with regard to the mahr. But he harmed her and put pressure on her, and he withheld her rights from her – without her deserving any such treatment – until she ransoms herself from him. Allah, may He be exalted, has forbidden such reprehensible conduct in the verse in which He says (interpretation of the meaning):
“O you who believe! You are forbidden to inherit women against their will, and you should not treat them with harshness, that you may take away part of the Mahr you have given them, unless they commit open illegal sexual intercourse. And live with them honourably. If you dislike them, it may be that you dislike a thing and Allah brings through it a great deal of good”
[an-Nisa’ 4:19]. 
Secondly: 
This talaaq that has been given by this husband is called a “talaaq in return for letting the husband off the mahr or some other obligation”; it is a kind of khula‘, as was explained in fatwa no. 198920 
The fuqaha’ differed as to the definition of a khula‘ or “talaaq in return for letting the husband off the mahr or some other obligation”. According to the majority of fuqaha’, it is an irrevocable talaaq. Al-Khurashi al-Maaliki said: An irrevocable talaaq can only come about using words that signal khula‘, “talaaq in return for letting the husband off the mahr or some other obligation”, ransom or talaaq with monetary payment. 
End quote from Sharh Mukhtasar Khaleel by al-Khurashi, 4/17 
One of the rulings on talaaq in return for payment is that the husband does not have the right to take the wife back during the ‘iddah. It says in al-Mawsoo‘ah al-Fiqhiyyah al-Kuwaitiyyah (22/108), when mentioning the conditions on taking the wife back: 
The fifth condition: that the talaaq should not be in return for compensation. If the talaaq is in return for compensation, then it is not valid to take the wife back, because in that case the talaaq is irrevocable, because the woman has ransomed herself from the husband by means of the financial compensation that she gave him, which ended this relationship, as in the case of khula‘ and talaaq in return for money. End quote. 
A number of scholars are of the view that khula‘ – along with “talaaq in return for letting the husband off the mahr or some other obligation” – is an annulment of the marriage and not a talaaq. This is the more correct view, as we have explained in fatwa no. 126444 
According to this more correct view, what happened with this man is an annulment and is not a talaaq at all. Hence he has no right to take back the wife from whom he separated by khula‘ during the ‘iddah. If he wants to marry her again, then he may do so, with a new marriage contract and mahr, if she herself agrees to that. 
Thirdly: 
Once it is proven that what happened comes under the heading of khula‘, then it does not matter whether it happened during menses or during a period of purity in which he had had intercourse with her, because the correct scholarly view is that there is nothing wrong with khula‘ during menses or during a period of purity in which intercourse took place. 
It says in al-Mughni by Ibn Qudaamah (7/324): There is nothing wrong with khula‘ during menses or during a period of purity in which he had intercourse with her, because the prohibition on talaaq during menses is because of the harm that will affect her as a result of making the ‘iddah longer, whereas khula‘ is aimed at removing the harm that affects her because of the husband's bad treatment and staying with someone whom she dislikes, which is worse than the harm caused by extending the length of the ‘iddah. Hence it is permissible to ward off the greater harm by means of the lesser. For this reason the Prophet (blessings and peace of Allah be upon him) did not ask the woman who was separated by means of khula‘ about her condition. Moreover, the harm caused by extending the length of the ‘iddah affects her, whereas khula‘ occurs at her request, which signals that she approves of it and indicates that it is most likely in her best interests. End quote. 
This issue has been discussed in detail in fatwa no. 186809 
Fourthly: 
With regard to the mahr that was given up by the wife, she does not have the right to change her mind, because she gave it up by choice after the husband offered to give her a talaaq in return for giving up the mahr, and she agreed to that. She could have refused and referred the matter to the shar‘i judge (qaadi) to make him divorce her (talaaq) because of the harm she was suffering. In that case she could have gotten her rights in full. 
Fifthly: 
With regard to the husband’s denial of this pregnancy and saying “I do not have a child from her,” this is reprehensible and is a false statement; there is a fear that he may come under the warning mentioned in the hadeeth in which the Prophet (blessings and peace of Allah be upon him) said: “Any man who denies his child whilst he is looking at him, will be obstructed from Allah, and Allah will disgrace him before all of creation, the first and the last of them.” Narrated by Abu Dawood, 2263; Ibn Maajah, 2743; classed as da‘eef by al-Albaani in as-Silsilah ad-Da‘eefah, 1427 
The fuqaha’ (may Allah have mercy on them) stated that any child born to a woman six months or more after the marriage contract, when it is possible that intercourse may have taken place thereafter, is to be attributed to the husband. 
The basic principle concerning that is the words of the Prophet (blessings and peace of Allah be upon him): “The child is for the (owner of the) bed and the fornicator gets nothing.”
Narrated by al-Bukhaari, 2053; Muslim, 1457. 
An-Nawawi (may Allah have mercy on him) said: As for the words of the Prophet (blessings and peace of Allah be upon him), “The child is for the (owner of the) bed”, what is meant is that if the man has a wife or a slave woman who has become his concubine, and she gives birth to a child within the timeframe during which it is feasible that it could be his child, the child is to be attributed to him and becomes his child; they may inherit from one another and all the other rulings stemming from birth are applicable to them. 
End quote from Sharh an-Nawawi ‘ala Muslim, 10/37 
With regard to what he has written, or even if she or her guardians wrote it, stating that he has no child from her, this is idle talk which is worthless and has no impact. If it so happened that he did not have a child from her at the time of writing, and her pregnancy was not known then, this does not mean that she did not bear him a child who was born after that, within the timeframe of feasibility, as mentioned above. It is not permissible for him to deny the child and it is not permissible either for her to conceal what Allah created in her womb from her husband, even if she hates him. 
And Allah knows best.