What is the wisdom behind legislating the Udhiyah?
The Muslim must know that through the Uḍḥiyah, he complies with the command of Allah, the Exalted, and performs a worship that brings him closer to Allah and distances him from the Fire.
The Uḍḥiyah holds profound wisdoms and noble meanings, including:
1-Reviving the Sunnah of Abraham, peace be upon him: He complied with Allah's command when He ordered him to slaughter his son Ishmael. Abraham, peace be upon him, succeeded in the test, and when he was about to slaughter his son, Allah sent down a ram to ransom Ishmael and commanded him to slaughter it instead. Allah, the Exalted, says {what means]: "And when he reached with him [the age of] exertion, he said, 'O my son, indeed I have seen in a dream that I [must] sacrifice you, so see what you think.' He said, 'O my father, do as you are commanded. You will find me, if Allah wills, of the steadfast.' And when they had both submitted and he put him down upon his forehead, We called to him, 'O Abraham, You have fulfilled the vision.' Indeed, We thus reward the doers of good. Indeed, this was the clear trial. And We ransomed him with a great sacrifice." [Al-Ṣāffāt/102-107]. Thus, the sacrificial offering became an enduring Sunnah until the Day of Judgment, and Muslims offer their wealth as sacrifices, drawing closer to Allah, the Exalted, reviving this great Sunnah.
2-Providing abundance for people on Eid day and the Days of Tashrīq: Among the wisdoms of the Uḍḥiyah is that the Muslim provides generously for his family, neighbors, relatives, and the poor during these days. The Muslim is encouraged to eat from his sacrifice, give charity from it to the poor, and gift from it to his wealthy neighbors. Thus, goodness spreads throughout the entire community. The Prophet, peace and blessings be upon him, said: "The Days of Tashrīq are days of eating and drinking." (Narrated by Muslim). Allah, the Exalted, says {what means}: "And the camels and cattle We have appointed for you as among the symbols of Allah; for you therein is good." [Al-Ḥajj/36]. This good includes the good of both this world and the Hereafter.
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.
What is the ruling on istinja' after urination, and is it done with water and soap or with water only?
Istinja' from urine and stool is obligatory. It is permissible to perform istinja' with water alone, or with toilet paper alone. The best way is with toilet paper then water. If one wishes to suffice with one of them, then water is better. And Allah the Almighty knows best.
What is the ruling on Friday Ghusl (ritual bath)?
Friday Ghusl is a confirmed Sunna (Prophetic tradition) even if a person wasn`t in a state of Janbah (ritual impurity), or physically unclean. However, one who doesn`t make Ghusl on Friday isn`t sinful for the Prophet (PBUH) said: "It is good for a Muslim to make ablution for Friday prayer, but it is better to make Ghusul. [At-Tirmithi & Abu Dawood].