A dowry is a transfer of parental property, gifts, or money at the marriage of a daughter. Dowry contrasts with the related concepts of bride price and dower. While bride price or bride service is a payment by the groom or his family to the bride's parents, dowry is the wealth transferred from the bride's family to the groom or his family, ostensibly for the bride.
In an Islamic marriage contract, it is necessary to stipulate a nuptial gift, or what we call sidāq or mahr, and it can be anything that has a value, like teaching to recite Quran, or a copy of a Quran, etc.
There are many benefits of having such a mahr and not demanding high mahr. There is a statement from the holy Prophet (s.a.w.) that says the best of the women of my ummah is she with the less mahr.
Mahr is not a sign of love, or a security bond, but only a gesture and token of truthfullness, and anything spiritual will of course carry many benefits in this dunya and in akhirah.
With prayers for your success.
It is possible that they pronounce the permanent marriage contract without mentioning the dowry i.e Mahr, so they can decide on what it will be with a mutual consent after the marriage.
But if the intercourse takes place before determining the dowry, the woman will be eligible to ask for the Mahrul Mesl i.e. a dowry which is usually given to this type of girl with these characteristics and family and social position.
The short answer is, it is prescribed by Quran and hadith.
A longer answer: In my view, part of the wisdom is that... sadly, the human being values what they pay for. Most people who pay $100 for a book will read it. Whereas most people who download books for free will never read them at all. A person who pays $1000 for a book will memorize every word in it!
While I do not want to suggest that a female is like a product that a man buys, it seems that, for some men, making a financial commitment towards a marriage is a way of demonstrating their overall commitment to and investment in it, and commitment is one of the necessary ingredients for a lasting marriage. (Whether that financial commitment be made in the way of mahr, gifts, money towards a home, paying towards the wedding, etc)
I am not saying all men are like this, it just seems to be part of how some people's minds operate. Of course two people can be genuinely committed to a marraige even if neither of them has anything financial to contribute.
Of course mahrs should be moderate and not extravagant and should be according to one's capability.
Also in reality although there are theoretically equal opportunities for men and women, women are still more disadvantaged and paid less in the workplace, and are in more vulnerable situations when they have children, especially if they end up as single mothers. So anything that can help add some "insurance" to the woman's situation is not a bad thing.
Yes it is permissible for her to gift her right , whether fully or partly, to her husband. She will get great reward for such gift to her husband and that amount will be a blessed money for the husband.( Sura 4, verse 4).
Dowry is the right of the woman in marriage. She has the right to ask for her dowry upfront or keep part of it for later. There is no condition at all to keep a deferred dowry if the woman does not want to defer it.