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.
What is the ruling on mentioning Allah`s name upon slaughtering an animal?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is a Sunnah (prophetic tradition) for the person slaughtering to say: "Bismillah" (In the Name of Allah), though the most complete and perfect phrasing is to say: "Bismillah ar-Rahman ar-Rahim" (In the Name of Allah, the Most Gracious, the Most Merciful). This is in accordance with the words of Allah Almighty: "So eat of that [meat] upon which the name of Allah has been mentioned." (Al-An'am/118)
However, pronouncing it is not an obligatory requirement (Wajib). If someone omits it—whether deliberately or out of forgetfulness—the meat remains entirely lawful (Halal) to eat, though purposely omitting it is considered disliked (Makruh). And Allah the Almighty Knows Best.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
What is the ruling of Islamic Law on saying the basmalah at the beginning of wudu`?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Saying the basmalah (Bismillāh) at the beginning of wuḍūʾ is a sunnah. It is stated in al-Minhāj al-Qawīm Sharḥ al-Muqaddimah al-Ḥaḍramiyyah (p. 27): “Its sunnahs include the siwāk, then the tasmiyah (saying Bismillāh), joined with the intention (niyyah), at the start of washing the hands.” And Allah Almighty knows best.