What are the conditions that must be met for the 'aqīqah to be valid?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The 'aqīqah is like the recommended uḍḥiyyah in terms of the type of animal, its age, and its required characteristics — since it is a recommended slaughter and therefore resembles the uḍḥiyyah in its rulings.
It is agreed upon without any difference of opinion that the 'aqīqah is not valid with any animal other than livestock (al-na'am — camels, cattle, and sheep). Likewise, an animal with a disqualifying defect — such as obvious illness, lameness, or blindness in one eye — does not suffice, as has previously been explained in the rulings of the uḍḥiyyah. And Allah Almighty knows best.
Is it permissible for one to give the Zakah (obligatory charity) to his indebted brother?
It is permissible for one to give the Zakah to his brother if he was indebted, or poor.
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.
Is it permissible for a woman to sit with her brothers-in-law?
It is impermissible for a woman to sit with her brothers-in-law, and it is also impermissible for a person to have a Khulwa (seclusion) with his sister-in-law.