Is washing the private part after urinating (Istinjaa`) a condition, and is it done with water and soap, or with water only?
Istinjaa` is obligatory for removing impurity, and it can be done with toilet paper, or a stone, or water. It is preferable to do Istinjaa` first with paper, or stone, or the like, then to wash the private part with water until making sure that the impurity has been removed. Using soap is not a condition, but there is no harm in doing so provided that it is washed off with water. It is also permissible to use either water, or paper if impurity is removed by any.
What is the ruling on someone who dies while having missed fasts?
● If a person dies before having the opportunity to make up the missed fasts—such as someone whose excuse (e.g., illness) persisted until their death—then no makeup fast (qada), fidyah, or sin applies to them.
● However, if they had the ability to make up the fasts but did not do so before passing away, the missed fasts must be compensated by giving a mudd of food for each missed day from their estate.
The Prophet ﷺ said: "Whoever dies while having a month’s fast due, one needy person should be fed per day on their behalf." [Narrated by At-Tirmidhi]
Additionally, a guardian (wali) may fast on their behalf, as the Prophet ﷺ said: "Whoever dies while having missed fasts, their guardian should fast on their behalf." [Narrated by Al-Bukhari and Muslim]
In another narration: "If they wish." This indicates that both feeding the needy and fasting on behalf of the deceased are permissible options.
Does passing wind, from the anus without a smell, invalidate prayer?
If a person is certain of having passed wind even if it was with no smell, then both his/her ablution and prayer are invalidated, thus he/she should remake ablution and re-perform prayer.
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.