the mahr is the gift from the groom, solely for the bride. It is hers, and hers alone to choose to save or to spend (do not forget zakah on it if you save

Muslimahsister, the mahr is for you. You take it, and then , you choose to give it to your grandmother. But as far as I know, you have to have it first,( otherwise you cannot classify it as a mahr but as any gift from husband, or you can make it a condition in your marriage contract if he agrees to it) and then give it away. It might be there are some difference of opinions that I am (of course) not aware of.
The mahr is a debt owed from the groom to the bride. Should he die, this debt should be included within all his other debts and still be paid.
If the bride chooses, she can remits part or the totally of the mahr to her husband, and there is no sin on him if he accepts as she is doing it from the goodness of her heart (sometimes, for example, some families exaggerate and ask for too much and the bride then remits what she consider over the top back to her husband.)
Let me find the Ayah in Surah Al Baqarah Inshallah about the permissibility for the husband to accept the mahr or part of it if the bride wants to give it back.
but sisters, please consider this :
the mahr is not something frivolous. It is A duty upon the groom to the bride (not to confuse it with the marriage contracts where the bride and grooms write down their conditions if accepted) and the groom HAS to pay it , regardless of what it is, and if he does not, he is transgressing the bondaries set by Allah (unless the wife has agreed to forgive the debt from the goodness of her heart).
Also, if you do not stipulte a time limit for the mahr to be paid, then you cannot say he does not want to pay it if he does not , unless he says it or acts it clearly.
In case of talaq and the mahr was not paid yet, the husband has to pay it at the time of divorce.
He cannot force his wife to ask for khula in order to get his mahr back nor push her to it .(some men do and they are transgressors)
If you give me just until tomorrow Inshallah, I will dig out for you all what I know and the daleel for the mahr.
I agree that learning the Quran should not be part of the mahr because a man should do it anyway.
It is why many times, many grooms do not realise how important honouring the mahr is. Honouring the mahr is honouring the bride (ie, paying it is honouring the bride)
I do believe the meaning and this importance of this bridal gift is getting lost because many people associate the mahr to a dowry... no, the dowry is something else and the mahr is for the bride.