Does swallowing a fly, road dust, or flour dust break the fast?
A fasting person does not break their fast if something enters their body cavity against their will, such as a fly, road dust, or flour dust.
Is it permissible to offer an Udhiyah on behalf of the deceased?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Offering a sacrifice (Udhiyah) on behalf of the deceased is permissible. This is the position of the Hanbalis [Kashshaf al-Qina’ by al-Bahuti (Vol.6/P.428)], and it was held by al-Abbadi of the Shafi’is [Bidayat al-Muhtaj by Ibn Qadi Shuhbah (Vol.4/P.358)]; it is also narrated from some Maliki and Hanafi scholars.
Abu Dawud included a chapter in his Sunan titled "Chapter: Offering the Sacrifice on Behalf of the Deceased," in which he narrated from Hanash, who said: "I saw Ali (may Allah be pleased with him) sacrificing two rams. I asked him, 'What is this?' He replied, 'The Messenger of Allah (peace be upon him) enjoined me to sacrifice on his behalf, so I am sacrificing on his behalf.'"
Abu Dawud also narrated from Jabir (may Allah be pleased with him) that the Prophet (peace be upon him) said: "O Allah, this is from You and for You, on behalf of Muhammad and his Ummah; in the name of Allah, and Allah is the Greatest," then he slaughtered it. It is well known that among the Ummah of Muhammad (peace be upon him) are those who have passed away, yet he (peace be upon him) dedicated it to his entire Ummah.
Furthermore, multiple Sharia texts have consistently indicated that the rewards of righteous deeds reach the deceased. This includes the permissibility of fasting on behalf of the deceased if they died owing fasts, as well as the permissibility of performing Hajj on their behalf, both of which are established in authentic Hadiths. Since the rewards for fasting—a physical act of worship—and Hajj—a physical and financial act of worship—reach the deceased, then the sacrifice (Udhiyah) is even more likely to reach them.
Moreover, the scholars have reached a consensus (Ijma') that the rewards of charities reach the deceased, and the Udhiyah is a form of charity and falls under its general category. Based on all of this, we hold the view that offering a sacrifice on behalf of the deceased is permissible. And Allah the Almighty knows best.
Should nail polish be removed before making ablution?
Yes, it should be removed because it prevents water from reaching the nails.
Is it permissible to purchase the Udhiyah on installments?
It is permissible to purchase the sacrificial animal (uḍḥiyyah) on installments or by borrowing its price. However, it is not recommended for the poor to do so, because they are not required to offer a sacrifice, and Allah does not burden a soul beyond its capacity. If doing so would lead to negligence in providing for one's dependents (nafaqah wājibah), then giving precedence to obligatory maintenance over borrowing to buy the sacrifice is appropriate. This is based on the saying of the Prophet (peace and blessings be upon him): "It is sufficient sin for a person to withhold food from those whom he is responsible to support."
Furthermore, the sacrifice is valid from one who has a debt, but it is preferable (awlā) to repay the debt first, especially if the debt is due immediately (ḥāll). And Allah Almighty knows best.
