Is it Sunnah to cook the meat of the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah not to distribute the meat of the 'aqīqah raw. Rather, it is recommended to cook it with something sweet — such as raisins or honey — as a good omen and expression of hope for the sweetness of the newborn's character and conduct throughout their life. And Allah Almighty knows best.
What is the ruling on a person in a state of major impurity (junub) or a menstruating woman (ha'id) reciting the Quran from memory?
It is not permissible for a menstruating woman, a postpartum woman, or a person in a state of major impurity to recite anything from the Quran, whether from memory, from the Quran, from a phone, or a computer. It is also not permissible for them to touch the Quran, based on what was reported from Ali bin Abi Talib that the Prophet (peace be upon him) was not prevented from anything regarding the Quran except major impurity (janabah). (Reported by al-Tirmidhi who said it is a hasan sahih hadith). Menstruation and postpartum bleeding are analogous to major impurity (janabah) as they are all major impurities (hadath akbar).
For those mentioned, it is permissible to mention Allah and supplicate even with verses from the Quran, provided they do not intend them as recitation of the Quran, but intend them as remembrance (dhikr) or supplication (du'a). 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 swearing an oath by the Prophet ﷺ, and does such an oath take effect according to Imām Aḥmad ibn Ḥanbal, requiring expiation upon its breach?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
Swearing an oath by a created being is disliked (makrūh) in our Shāfiʿī school. Shaykh al-Islām Imām al-Nawawī, may Allah have mercy upon him, states: "Swearing by a created being is disliked — such as swearing by the Prophet, the Kaʿbah, Jibrīl, the Companions, or the Prophet's family. Al-Shāfiʿī, may Allah have mercy upon him, said: 'I fear that swearing by other than Allah the Almighty may constitute an act of disobedience.' The scholars of the school explained this to mean: that is, something forbidden and sinful — indicating that he had some hesitation in the matter. Al-Imām stated: the established position of the school is that it is categorically not forbidden, but rather disliked. Furthermore, whoever swears by a created being, his oath does not take effect and no expiation (kaffārah) is required if he breaks it." [Rawḍat al-Ṭālibīn wa ʿUmdat al-Muftīn, Vol. 11/P.6]
According to the Ḥanbalī school, however, expiation becomes obligatory upon one who swears by our master the Prophet ﷺ and then breaks his oath. Imām al-Bahūtī al-Ḥanbalī, may Allah have mercy upon him, states: "No expiation is required for swearing by other than Allah the Almighty, even if the oath is broken — because expiation was made obligatory for swearing by Allah and His attributes, out of reverence for His names, and nothing else is equal to Him in this regard... except in the case of swearing by our Prophet Muḥammad ﷺ, for expiation becomes obligatory when one swears by him and then breaks the oath. This was explicitly stated in the narration of Abū Ṭālib, because he is one of the two conditions of the two testimonies of faith by which a disbeliever becomes a Muslim. Ibn ʿAqīl held the view that swearing by any of the other prophets, peace and blessings be upon them all, carries the same ruling." [Sharḥ Muntahā al-Irādāt, Vol. 3/P.441]. And Allah the Almighty knows best.