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.
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.
Is it permissible for a woman to shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram (unmarriageable) of hers.
Does burping (belching) break the fast?
Burping does not break the fast. However, if anything comes up from the stomach, the fasting person must spit it out and rinse their mouth to cleanse it.
If they swallow it intentionally while being able to expel it, or if they swallow their saliva before rinsing their mouth, their fast is invalidated.