What is the ruling of Islamic Law on a latecomer who joins the imam during the standing position of the first rakʿah but was unable to complete the recitation of al-Fātiḥah?
All praise is due to Allah, and peace and blessings be upon our master the Messenger of Allah ﷺ.
If a latecomer (masbuq) joins the prayer while the imam is standing, he should begin reciting Sūrat al-Fātiḥah immediately, without pausing to recite the opening supplication (duʿāʾ al-istiftāḥ) or the seeking of refuge (taʿawwudh). If the imam bows before he completes al-Fātiḥah, he follows the imam into the bow and leaves whatever remains of al-Fātiḥah — the imam bears it on his behalf.
It is stated in ʿUmdat al-Sālik (p. 47): "If a latecomer finds the imam standing and is confident that he has enough time to recite the taʿawwudh and al-Fātiḥah in full, he may do so. If he is uncertain, he should neither recite the opening supplication nor the taʿawwudh, but rather begin directly with al-Fātiḥah. If the imam bows before he completes it, he follows him into the bow — provided he had not already begun the opening supplication or the taʿawwudh. If he had begun either of them, he continues reciting al-Fātiḥah for as long as he spent on them." And Allah the Almighty knows best.
What is the ruling on performing ablution (wudu') and ritual bathing (ghusl) with Zamzam water?
It is permissible to perform ablution and ritual bathing with Zamzam water. However, scholars disliked using it for cleaning after relieving oneself (istinja'). And Allah the Almighty knows best.
Is it permissible to trim the eyebrows if they become excessively long?
Trimming eyebrows is permissible if they become excessively long, yet; it is reprehensible. And Allah Knows Best.
Is it permissible to pay the Fitr Zakah (obligatory charity) of Ramadan on behalf of a dead person?
The Fitr Zakah of Ramadhaan isn`t due on one who had passed away before the sunset of the last day of Ramadan. And Allah Knows Best.