What is the ruling on using eye drops while fasting?
Eye drops do not break the fast, even if the taste reaches the throat, because the eye is not an open passage to the body cavity.
What is the ruling on omitting the prostration of recitation?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The prostration of recitation (Sajdat al-Tilawah) is a Sunnah for both the reader and the listener. There is no sin in omitting it, though doing so results in missing a great reward. Muslim narrated from Abu Hurairah (may Allah be pleased with him) that the Prophet ﷺ said: 'When the son of Adam recites a verse of prostration and prostrates, Shaytan withdraws weeping, saying: "Woe to him! (and in the narration of Abu Kuraib: "Woe to me!") The son of Adam was commanded to prostrate and he prostrated, so Paradise is his; and I was commanded to prostrate and I refused, so the Fire is mine."'
Sheikh al-Islam Imam al-Nawawi (may Allah have mercy on him) stated: 'It is recommended to prostrate immediately after reciting or hearing a verse of prostration. If one delays it but the interval is short, he may still prostrate. However, if the interval is long, the opportunity is missed.' [Rawdat al-Talibin Vol.1/P.323].
Furthermore, the prostration of recitation becomes obligatory (Wajib) in congregational prayer if the Imam prostrates, out of the necessity of following him. And Allah the Exalted knows best.
Does sacrificing one sheep avail for the entire household?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
Offering an Udhiyah (sacrificial offering) is a communal Sunnah (Sunnah Kifayah) for members of the same household, provided that their financial maintenance is undertaken by a single provider. By "communal Sunnah," we mean that when one person performs it, the religious recommendation is fulfilled on behalf of the entire household, though the spiritual reward itself belongs uniquely to the one who offered it.
Therefore, if any member of the household performs the sacrifice—even if it is someone who is not legally responsible for the household's expenses, such as the wife or one of the children—the recommendation is fulfilled for everyone in that home. However, the reward does not automatically extend to the other members unless the person offering the sacrifice explicitly intends to share the reward with them—similar to how performing a funeral prayer (Janazah) fulfills the communal obligation for everyone, yet the specific reward is earned by those who actually prayed.
Additionally, a single sacrifice is sufficient for a man who is married to more than one wife. And Allah the Almighty Knows Best.
What is the ruling on praying behind a disliked Imam?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is mildly disliked (Makruh Tanzihan) for a man—other than the permanent designated Imam (Imam Ratib)—to lead people in prayer if the majority of them dislike him due to a religiously blameworthy reason (Amr Madhmum Shar‘an), such as associating with open sinners (Fussaq), for example. As for the followers who dislike him, the prayer behind him is not disliked for them. However, if less than the majority dislike him, or if the majority dislike him for a reason that is not religiously blameworthy, then his leading the prayer is not disliked. And Allah the Exalted knows best.