Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.
What are the categories of the livestock permissible for Udhiyah?
Praise be to Allah, and peace and blessings be upon the Messenger of Allah.
The Udhiyah (sacrificial offering) is not valid except from Al-An’am (livestock), which are: camels, cattle, and sheep/goats. Allah the Almighty says {what means}: "And for every nation We have appointed a rite [of sacrifice] that they may mention the name of Allah over what He has provided for them of beast-livestock." [Al-Hajj/34]
The best of them are camels, then cattle, then sheep/goats. A sacrifice of a camel or a cow avails for seven people. Jabir (may Allah be pleased with him) said: "We sacrificed with the Messenger of Allah (peace and blessings be upon him) in the year of Al-Hudaybiyah a camel for seven and a cow for seven." [Narrated by Muslim] And Allah the Almighty knows best.
What are the Sunnahs and etiquettes recommended for the person offering the Udhiyah?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
There are several Sunnahs and etiquettes that are recommended for the person offering the Udhiyah (sacrifice) to observe, including:
1-It is Sunnah for the one intending to sacrifice to refrain from removing any hair or nails once the first ten days of Dhul-Hijjah begin. The Prophet (peace be upon him) said: "When the ten days [of Dhul-Hijjah] begin and one of you intends to offer a sacrifice, let him not touch [remove] any of his hair or skin." [Narrated by Muslim]. If someone does remove any hair or nails, they have not committed a sin, and their sacrifice remains valid.
2-It is recommended for the person offering the sacrifice to slaughter the animal themselves. If they are unable to do so, they should witness the slaughter. The Messenger of Allah (peace be upon him) said to Fatimah (may Allah be pleased with her): "Stand and witness your sacrifice, for indeed, with its first drop of blood, your [previous sins] are forgiven." [Narrated by Al-Bayhaqi and Al-Tabarani; it is classified as a weak (da'if) hadith].
3-The animal should be positioned to face the Qiblah at the time of slaughter, as the Qiblah is the most noble of directions.
4-The slaughterer should say: "Bismillah ar-Rahman ar-Rahim" (In the name of Allah, the Entirely Merciful, the Especially Merciful). Even if one forgets to say it, the sacrifice is still permissible to eat. Allah (Blessed and Exalted be He) says: "So eat of that [meat] upon which the name of Allah has been mentioned" [Al-An'am: 118]. It is also recommended to send blessings upon the Messenger of Allah (peace be upon him) and to say "Allahu Akbar" (Allah is the Greatest) after the Tasmiyah.
5-The slaughterer should pray for the sacrifice to be accepted by saying: "O Allah, this is from You and for You, so please accept it from me." (Allahumma hadhihi minka wa ilayka, fataqabbal minni). And Allah the Almighty knows best.
Is it permissible for a menstruating woman to recite the Quran from the computer without actually touching the Quran?
It is not permissible for a woman in her menstrual period, or in postpartum to recite the Quran, even by heart, or without actually touching the Quran itself. Recitation itself is what is prohibited on her regardless of how it is done. However, it is permissible for her to surf through the Quran by her eyes, or to recall it in her head without uttering the words. There is no harm for her to look at the Quranic verses on the computer without touching it, or to utter the words as this is called looking not reciting or reading.