How is the Night Prayer (Qiyam al-Layl) performed?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Night Prayer (Qiyam al-Layl) refers to the voluntary (Nafl) prayers performed by a person after the Sunset prayer (Maghrib) and before the Dawn prayer (Fajr). As for Tahajjud, it is the voluntary prayer performed during the night specifically after waking up from sleep, as an act of devotion to Allah. Allah the Exalted says {what means}: "And from [part of] the night, pray with it [i.e., recitation] as additional worship for you." (Al-Isra/79). Thus, in terms of reward,Tahajjud is superior to voluntary prayers performed before sleeping. And Allah the Exalted knows best.
Is "Bismillāhir-Raḥmānir-Raḥīm" a verse (āyah) of Sūrat al-Fātiḥah?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
The basmalah ("Bismillāhir-Raḥmānir-Raḥīm") is, without question, a verse (āyah) of the Noble Qur'an, and according to the Shāfi‘ī school, it constitutes a verse of Sūrat al-Fātiḥah itself. Al-Fātiḥah is a pillar of the prayer, and it consists of seven verses, of which the basmalah is one. The prayer is not valid without reciting it, nor if any of the Fātiḥah's verses, letters, or shaddahs (consonant doublings) are omitted or altered.
It is stated in Bushrā al-Karīm (Vol.1/p.202), from the books of the Shāfi‘ī school: "The basmalah is a complete verse of it [i.e., of al-Fātiḥah] in practice — and it suffices, for this to be established, that there be strong probability (ẓann), especially given that this has come very close to certainty (yaqīn), due to the consensus (ijmā‘) of the Companions on its inclusion in the muṣ-ḥaf [the written text of the Qur'an] in its script, despite their extreme diligence in excluding from it anything that did not genuinely belong to it — to the point that they even [carefully regulated] its diacritical dots and vowel markings." And Allah, the Exalted, knows best.
What is the ruling on performing the obligatory prayer immediately after the commencement of the Adhan?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to perform the obligatory prayer (Fard) immediately after the commencement of the prayer time. As the Adhan (call to prayer) serves as a sign that the time has entered, it is permissible to pray as soon as the Adhan begins. However, it is recommended (Mustahabb) for whoever hears the Mu’adhin to repeat what he says. If the Adhan occurs while one is already engaged in prayer, he should repeat the words of the Mu’adhin after he has completed his prayer. And Allah the Exalted 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.