What is the ruling on seeking forgiveness between the two Friday sermons?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is permissible to seek forgiveness (Istighfar) between the two Friday sermons (Khutbatayn), and it does not invalidate the Friday prayer (Salat al-Jumu'ah). Furthermore, it is not disliked (Makruh) to speak before the sermon, after it, or between the two sermons. Likewise, it is not disliked for one who enters during the sermon to speak if there is a need for it, provided he has not yet taken a place and settled into it. And Allah the Exalted knows best.
Is it a condition for I‘tikaf to be performed in the mosque?
Yes, for I‘tikaf to be valid, it must be performed in a mosque. I‘tikaf is not valid if performed at home or elsewhere.
Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
Which is superior: The Udhiyah (Sacrificial Offering) or giving its value in Charity (Sadaqah)?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
Performing the Udhiyah (sacrificial offering) is superior to giving its monetary value as charity (Sadaqah) to the poor and needy. This is because the Udhiyah is one of the manifest symbols of the religion. Allah the Almighty says {what means}: "That [is so]. And whoever honors the symbols of Allah - indeed, it is from the piety of hearts." [Al-Hajj/32]. This is further supported by the explicit and authentic Hadiths regarding its virtue, and the fact that the Prophet ﷺ and the Rightly Guided Caliphs after him consistently performed the sacrifice. And Allah the Exalted knows best.