Is it a condition for the mosque where I‘tikaf is performed to hold Jumu‘ah prayers?
No, it is not a condition for the mosque where I‘tikaf is performed to have Jumu‘ah prayers. However, if a person vows to perform continuous I‘tikaf, and Jumu‘ah occurs within that period, then they must observe I‘tikaf in a mosque where Jumu‘ah is held so that they do not break their continuity by leaving for Friday prayer.
What should one who doubts the validity of his ablution and performs it frequently do?
Whoever doubts the validity of his ablution after finishing it, his ablution is valid. This is because the default is the validity of the ablution, and because doubt after finishing an act of worship doesn`t affect its validity. And Allah the Almighty knows best.
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.
What is the ruling on entering the toilet with something containing the remembrance of Allah?
It is disliked to bring into the toilet anything containing a reminder of Allah, such as the Quran or a sacred name (like the name of Allah, Muhammad, etc.). If one does that, etiquette dictates that he conceal it so it is not visible while entering the toilet, either by cupping his hands over it, putting it in his pocket, or inside his shirt if it is a necklace or ring. And Allah the Almighty knows best.