What is the ruling on the Friday bath (ghusl al-Jumu'ah)?
The Friday bath is a confirmed Sunnah, based on the Prophet's (peace be upon him) saying: "Whoever performs ablution on Friday, it is good for him, and whoever takes a bath, bathing is better." (Reported by Abu Dawud and al-Tirmidhi who said it is a hasan hadith). It is recommended for everyone who wants to attend the Friday prayer, even if they are not from its usual attendees, like a woman or a boy.
If a person is in a state of major impurity (junub) and intends by his bath both lifting the major impurity and the Friday bath, he achieves both. If he intends only one of them, he achieves only what he intended. And Allah the Almighty knows best.
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 meaning of the Prophetic statement that a boy is held in pledge (murtahan) for his 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The first interpretation: That if the boy dies in infancy without an 'aqīqah having been performed on his behalf, he will not intercede for his parents on the Day of Resurrection. This is the position of Imam Aḥmad ibn Ḥanbal, and Imam al-Khaṭṭābī concurred with him, stating: "The finest of what has been said regarding this matter is the position adopted by Aḥmad ibn Ḥanbal." — [Fatḥ al-Bārī by Ibn Ḥajar,{Vol.9/P.594]
The second interpretation: That the child is likened to a pledged object (marhūn) — one from which full benefit and enjoyment cannot be derived until it is redeemed. A blessing is only made complete upon the one blessed when they fulfil the obligation of gratitude (shukr), and the prescribed expression of gratitude for this particular blessing is what the Prophet ﷺ established as Sunnah — namely, the slaughtering of the 'aqīqah on behalf of the newborn as an act of thankfulness to Allah the Almighty and as a supplication for the wellbeing and safety of the child. This is the position of Mullā 'Alī al-Qārī. See: [Mirqāt al-Mafātīḥ Sharḥ Mishkāt al-Maṣābīḥ, Vol.7/P.2688]
And Allah Almighty knows best.
Do pregnant and breastfeeding women have to fast?
Pregnant and breastfeeding women are required to fast. However, if fasting causes them harm or unusual hardship, they may break their fast but must make up for the missed days later.
If they break their fast solely out of fear for the fetus or the child, then they must both make up the fast and give fidyah (feeding a needy person for each missed day), as the benefit of breaking the fast was only for the child.