Is the Saying "Whatever is Taken by the Sword of Shyness is Forbidden" an Authentic Ḥadīth?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
The saying "Whatever is taken by the sword of shyness is forbidden" is not an authentic ḥadīth, though its underlying meaning is sound. The established sharʿī principle is that a Muslim's wealth is not lawful for anyone to take except with his wholehearted consent, as Allah the Almighty says {what means}: "O you who have believed, do not consume one another's wealth unjustly, but only [in lawful] business by mutual consent." [Al-Nisā/ 29] And the Messenger of Allah ﷺ said: "Listen to me and you will live well: do not wrong others, do not wrong others, do not wrong others. Indeed, a man's wealth is not lawful except with his full, willing consent." (Reported by Aḥmad in his Musnad.) Whatever is taken through the pressure of shyness or social embarrassment runs directly counter to genuine, wholehearted consent.
The jurists have explicitly stated that whatever is taken by means of the "sword of shyness" carries the same ruling as that which is taken by coercion — it must be returned to its rightful owner.
Ibn Ḥajar al-Haytamī, may Allah have mercy upon him, states in al-Fatāwā al-Kubrā (Vol.3/P.30): "Do you not see the reported scholarly consensus that whoever has something taken from him purely out of shyness, without his genuine consent, does not pass ownership of it to the one who took it? They reasoned that this constitutes a form of coercion through the 'sword of shyness,' comparable to coercion at the point of an actual sword. Indeed, many people would rather submit to the literal sword and endure the pain of its wound than submit to this first kind of coercion, out of fear for their dignity and standing — which people of sound judgment hold dear and guard most fiercely." And Allah the Almighty knows best.
Is it permissible for a mother to stop her daughter from marrying a respectable, God-fearing young man because he isn't college/university graduate?
It is impermissible to deny a woman her right in getting married except for a lawful reason, and problems can be solved through dialogue. However, she can file a suit in order for the judge to have a say in that regard.
What is the ruling on vomiting, and does it invalidate ablution?
Vomit is impure (najis). Its exit is not considered one of the nullifiers of ablution. However, the mouth must be washed and purified from it, and any that gets on clothing or the body must be washed for prayer, because prayer is not valid with impurity present on the body or clothing. And Allah the Almighty knows best.
What is the ruling on the ablution of one who touches women prohibited to him by a temporary prohibition (mahram bi-hurma mu'aqqata)?
It is prohibited to touch a woman who is temporarily prohibited (meaning one whom it becomes permissible to marry after the impediment is removed, such as the wife of a brother or paternal uncle). Touching her without a barrier invalidates ablution. And Allah the Almighty knows best.