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.
Does using a respiratory device (oxygen) affect the validity of fasting?
Using an oxygen device does not affect the validity of fasting because oxygen is a gas with no physical substance.
However, if medicinal substances with physical particles are added to the oxygen, its use would break the fast, as it enters the body cavity through a natural passage.
What is the ruling of Islamic Law on performing ṭawāf on the upper floors of al-Masjid al-Ḥarām, given that these floors are situated above the level of the Ka‘bah, and that the one performing ṭawāf passes by certain barriers, such as pillars and walls, during the ṭawāf?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
There are numerous conditions for the validity of ṭawāf (circumambulation around the Ka‘bah), among them: that the one performing ṭawāf be inside the Masjid (al-Masjid al-Ḥarām) — even if the Masjid has been expanded, so long as the expansion does not extend out into al-ḥill (land outside the sanctuary boundaries). It suffices for the ṭawāf to be performed in the airspace of the Masjid or upon its roof, even if that roof is higher than the Ka‘bah (the House) itself.
Al-Khaṭīb ash-Shirbīnī (raḥimahullāh) said: "It [ṭawāf] is valid upon the roof of the Masjid, even if the roof of the Masjid is higher than the House — just as prayer [is valid] upon Mount Abū Qubays despite its elevation above the House. This is the relied-upon (mu‘tamad) position." [Mughnī al-Muḥtāj, Vol.2/P.246]
The validity of the tawaf isn`t affected by the pillars and walls standing between the one performing ṭawāf and the House, since the condition is only that the person performing ṭawāf be within the boundaries of the Masjid and outside the boundaries of the Ka‘bah itself.
Ash-Shirbīnī said: "It [ṭawāf] is valid inside the Masjid, even if it is expanded, and even if a barrier stands between the one circumambulating and the House — such as a water-supply structure (saqāyah) or pillars (sawārī)." [Mughnī al-Muḥtāj, Vol. 2/P.246]
Accordingly, ṭawāf performed on the upper floors is valid, and the presence of barriers such as pillars and walls during the ṭawāf does not affect its validity. And Allah, the Exalted, knows best.
When is Istikharah (guidance prayer) Du`a offered, before Tasleem (saying asalamu alikum warahmatu allah wa barakatuh to end the prayer), or after it?
It is offered after Tasleem.