Who is the one required to slaughter the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah for the 'aqīqah to be slaughtered by the guardian of the newborn — the one upon whom the child's financial maintenance is obligatory — provided he is financially capable of doing so. 'Alī ibn Abī Ṭālib, may Allah be pleased with him, reported: "The Messenger of Allah ﷺ performed the 'aqīqah for al-Ḥasan with one sheep." — Narrated by al-Tirmidhī. And Allah Almighty knows best.
Is it permissible for a woman to observe voluntary fast without her husband's permission?
A woman is not permitted to observe a voluntary fast (nafl) while her husband is present without his permission.
My brother works in a conventional bank and gave me one of the gifts distributed to bank employees — what is the ruling on accepting it?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
There is no objection to accepting such gifts, because the physical gift itself does not carry any inherent prohibition — unlike stolen property. Sin does not transfer or extend to the one who receives the gift, for Allah the Almighty says {what means}: "And no bearer of burdens shall bear the burden of another." [Al-Anʿām/164]
The evidence for this is that the Prophet ﷺ himself ate from the food of the Jews, conducted transactions with them, and purchased from them — and it is well known that their wealth was intermingled with ribā. Similarly, the wealth of conventional ribā-based banks is a mixture of the lawful and the unlawful.
Ibn Ḥajar al-Haytamī, may Allah have mercy upon him, states: "It is not forbidden to transact with one whose wealth is predominantly unlawful, nor to eat from it — as al-Nawawī affirmed in al-Majmūʿ." [Tuḥfat al-Muḥtāj, Vol. 9/P.389] And Allah the Almighty knows best.
Is it permissible for a woman to ride a taxi without a Mahram (unmarriageable kin) in order to attend a gathering of knowledge?
It is permissible for the woman to pursue the knowledge that she needs in order to perform the religious requirements due on her if there was nobody to teach her at home, provided that there is no temptation involved. Moreover, we don`t recommend that she rides a taxi without a Mahram although such an act isn`t from the forbidden Khulwah (seclusion).