Is it permissible for the girl and her family to demand half of the Mahr (dowry), or compensation for psychological harm, after the young man had handed over the agreed-upon Mahr to finalize the contract, but then the contract was not concluded the following day for unknown reasons?

The engagement and agreement on the mahr (dowry) and setting a date for the marriage contract do not constitute a legal (shar'i) contract. If there has been no offer (ijab) from the woman's guardian and acceptance (qabul) from the man in the presence of two witnesses, then the contract has not been concluded. It is merely an engagement, and its dissolution does not entail entitlement to any part of the mahr or compensation. The mahr is a right of the man to be handed over to him when he requests it, because the woman is not entitled to it except through the marriage contract. Whatever is sent for the mahr is returned in its exact form, if it is still extant, if the marriage does not take place.