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 to slaughter a single sheep with the combined intention of both the uḍḥiyyah and the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
According to Shaykh al-Islām Imam Ibn Ḥajar al-Haytamī, it is not permissible to combine the intention of the uḍḥiyyah and the 'aqīqah in a single animal, as each of the two has a distinct and separate cause that differs from the other.
However, Shaykh al-Islām Imam al-Ramlī permitted the combining of both intentions in a single animal — and this position offers a degree of latitude and ease. And Allah Almighty knows best.
What is the ruling on undoing braids of hair during the ritual bath?
It is not a condition to undo braids if the water reaches all the hair and penetrates to its base. If the water does not reach except by undoing the braids, then it is obligatory to undo them for the water to reach. And Allah the Almighty knows best.
Is it valid to perform Tarawih prayer at home?
It is Sunnah for a Muslim to perform Tarawih prayer in congregation at the mosque.
However, if a person sometimes prays it at home in congregation with their family for a valid reason, there is no harm in doing so.