Is the marriage of a young girl to her paternal cousin valid if the father says: "I have married my daughter to your son" while they are both young, and is it obligatory to give the dowry to the girl if they both grow up and the boy does not consent to her?

It is permissible for a father to marry off his young son or daughter. If the marriage is for a male, it should be with a dower (mahr) equivalent to that given to his peers (mahr al-mithl). If it is for a female, it should be to a righteous man. The jurists (fuqaha') have agreed on the permissibility of a father marrying off his young daughter without her permission. The evidence for this is Allah's Almighty saying: "And as for those of your women who have despaired of menstruation, if you doubt, then their period is three months, and those who have not menstruated." In the scenario inquired about, if there was an offer and acceptance from both guardians in the presence of two witnesses, there are two opinions among the Shafi'is regarding the validity of the marriage. According to the opinion that affirms its validity, the young girl is entitled to a dower equivalent to that given to her peers. If the son grows up and is not pleased with his wife, then if he consummates the marriage, she is entitled to the entire dower. If he divorces her before consummation, she is entitled to half of her dower; this is based on Allah Almighty's saying: "And if you divorce them before you have touched them and you have already specified for them an obligation [dower], then for them is half of what you specified."