What are the Sharia consequences when the sacrificial time for the uḍḥiyyah comes to an end?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
If the sun sets on the final day of Tashriq (the 13th of Dhu al-Hijjah) and the Udhiyah (sacrificial animal) has not yet been slaughtered, its designated time has expired. Should a person slaughter it after this point, it will not be counted as an Udhiyah.
However, if the sacrifice was a vowed one (Mandhurah), they are strictly obligated to slaughter it as a makeup act (Qada’), and its meat must be distributed entirely according to the rules governing vowed sacrifices.
It is stated in Bushra al-Karim (p. 702): "If one slaughters after sunset on the final day [of Tashriq]... it does not count as an Udhiyah, unless it was a vowed sacrifice, in which case it is fulfilled as a makeup act (Qada’)." 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.
How is night prayer offered?
Night prayer is offering voluntary prayer after Maghrib and before Fajr (Dawn). As for Tahajjud, it is offering voluntary prayer at night after waking up voluntarily, and for Allah`s sake as He Says (What means): "And pray in the small watches of the morning: (it would be) an additional prayer (or spiritual profit) for thee: soon will thy Lord raise thee to a Station of Praise and Glory!" [Al-Isra`/79]. Offering Tahajjud is better than offering voluntary prayer before going to bed.
What is incumbent upon the one offering the sacrifice if, after slaughtering the animal, they discover that one of its internal organs is damaged or diseased?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The presence of disease or defect in the internal organs of a sheep does not affect the validity of the sacrificial animal, unless the disease leads to the animal becoming emaciated and its meat becoming corrupted.
It is stated in al-Iqnā' (2/590) by Imam al-Shirbīnī: "The third disqualifying condition: an animal with a manifest illness — meaning one whose illness visibly results in emaciation and corruption of its meat. However, if the illness is minor and does not produce such effects, it doesn`t affect the validity of the sacrificed animal." And Allah Almighty knows best.