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 the fasting of a woman who has reached menopause if menstrual blood flows?
If a woman reaches the age of menopause (which is usually sixty-two) and her menses have ceased, then she sees blood after that, and its duration is not less than a day and a night (24 hours), it is menstruation (hayd). If it is less than a day and a night, she is considered as having non-menstrual vaginal bleeding (mustahada), so she fasts and prays. However, she must perform ablution for every obligatory prayer after its time enters, pray immediately, and be treated as a person with a continuous condition. There is no specific end limit for a woman's menstruation; it is possible as long as the woman is alive. And Allah the Almighty knows best.
Is it permissible for a woman to wear underwear while performing Umrah?
Yes, it is permissible for a woman to wear underwear while performing Umrah. This is because she should keep her regular clothes that cover all her body while being in a state of Ihram (ritual consecration) for Hajj, or Umrah. However, she should uncover her face and hands, but it is permissible for her to let her head-covering garment drape from her head down over her face when non-Mahram (i.e., marriageable) men pass by her. And Allah Knows Best.
What are the Sunnahs and etiquettes recommended for the person offering the Udhiyah?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
There are several Sunnahs (prophetic traditions) and etiquettes that are highly recommended for the person offering an Udhiyah (sacrificial offering) to observe:
First:
It is a Sunnah for anyone intending to offer a sacrifice to refrain from cutting or removing any of their hair or nails once the first ten days of Dhul-Hijjah begin. This is based on the statement of the Prophet (peace and blessings be upon him): "When the ten days [of Dhul-Hijjah] begin and one of you intends to offer a sacrifice, let him not touch any of his hair or skin." (Narrated by Muslim)
However, if someone does happen to remove any hair or nails, they have not committed a sin, and their sacrifice remains perfectly valid.
Second:
The person offering the sacrifice should ideally slaughter the animal themselves. If they are unable to do so, they should at least witness its slaughter. This is drawn from what our Master, the Messenger of Allah (peace and blessings be upon him), said to Lady Fatima (may Allah be pleased with her):
"Stand up and witness your sacrifice, for with its very first drop of blood, every sin you have committed will be forgiven." (Narrated by Al-Tabarani in his Mu'jam, Al-Hakim in his Mustadrak, and Al-Bayhaqi in his Sunan)
Third:
Face the Qiblah (the direction of prayer) at the time of slaughtering, as the Qiblah is the most honorable of directions.
Fourth:
Pronounce the name of Allah (Tasmiyah) at the moment of slaughter by saying: "Bismillah ar-Rahman ar-Rahim" (In the Name of Allah, the Most Gracious, the Most Merciful). Even if one forgets to say it, the meat remains lawful (Halal) to eat. Allah, Blessed and Exalted is He, says:
"So eat of that [meat] upon which the name of Allah has been mentioned." (Al-An'am: 118)
It is also recommended to send blessings upon our Master, the Messenger of Allah (peace and blessings be upon him), and to follow it with the Takbeer (Allahu Akbar).
Fifth:
Supplicate for acceptance by saying: "Allahumma hadhihi minka wa ilayka, fa-taqabbal minni" (O Allah, this is a blessing from You and is offered back to You, so please accept it from me). This means: This sacrifice is a blessing that originated from You, and I am offering it to draw closer to You. And Allah the Almighty Knows Best.