Is the son of the sister by breastfeeding considered a mahram in Islam, and is it permissible for a woman to go for Hajj with him?
Everything that is prohibited by lineage is also prohibited by breastfeeding. The son of the sister by breastfeeding is considered a mahram, just like the son of the sister by blood, and it is permissible for him to be her mahram for Hajj and Umrah. And Allah Knows Best.
When is Istikharah (guidance prayer) Du`a offered, before Tasleem (saying asalamu alikum warahmatu allah wa barakatuh to end the prayer), or after it?
It is offered after Tasleem.
What is the ruling on someone who intends to travel to Jeddah for two days — must he enter iḥrām from his place of residence [i.e., before passing the mīqāt]?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
Whoever is firmly resolved in his intention to perform ‘Umrah before setting out on his journey, it is unlawful for him to pass beyond his country's mīqāt (the designated boundary point for entering the state of iḥrām) without entering iḥrām. If he does so, he must return to the mīqāt to enter iḥrām from there, provided he has not yet begun performing the rites of the pilgrimage — in which case nothing further is required of him. However, if he does not return, he must offer a sacrifice (sheep/goat), to be distributed among the poor of the Ḥaram.
It is stated in at-Tanbīh fī al-Fiqh ash-Shāfi‘ī (p. 71): "Whoever, intending to perform the rites [of ‘Umrah or Ḥajj], passes beyond the mīqāt and enters iḥrām only after it, must offer a sacrifice. However, if he returns to the mīqāt before beginning any of the rites, the sacrifice is waived from him."
Accordingly: whoever is firmly resolved in his intention to perform ‘Umrah before setting out on his journey must enter iḥrām at the mīqāt of his country through which he passes. Whoever fails to do so, passing beyond the mīqāt without entering iḥrām, and does not return, has sinned, and must repent to Allah, the Exalted; and he must offer a sacrifice, to be distributed among the poor of the Ḥaram. And Allah, the Exalted, knows best.
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.