What is the ruling on performing the Istikhara prayer after the Witr paryer?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The Istikhara prayer (Prayer for seeking guidance) is a Sunnah. It consists of two units (rak’ahs) performed outside of the obligatory prayers, after which the person supplicates with the traditionally narrated du’a. It is permissible to perform it before or after the Witr prayer, as the Istikhara prayer is recommended at all times except during the disliked times—the periods in which prayer is prohibited. This is because its specific reason (the Istikhara and supplication) occurs after the prayer itself, and any prayer with a subsequent reason is not permitted during the prohibited times. It should be noted that the two rak’ahs of Istikhara are not fulfilled by performing only one rak’ah, nor by a prostration of recitation (Sajdat al-Tilawah), nor by a funeral prayer (Janazah). And Allah the Exalted knows best.
What is the wisdom and purpose behind the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
Among the wisdoms and purposes of the 'aqīqah are the following:
First: Expressing joy at the blessing of Allah the Almighty, rejoicing in the newborn, and giving thanks to Allah for this gift. Allah the Almighty says (what means): "If you are grateful, I will surely increase you." [Ibrāhīm/7]
Second: Proclaiming and publicising the lineage (nasab) of the child — and the 'aqīqah serves as an appropriate and fitting means to this end.
Third: Gladdening the hearts of family members, relatives, friends, and the poor by gathering them together over a shared meal. And Allah Almighty knows best.
Does buying meat equivalent to the weight of a slaughtered animal and distributing it avail for an udhiyah?
Praise be to Allah, and prayers and peace be upon our Master the Messenger of Allah.
That does not avail for the sacrifice (uḍḥiyyah), because the sacrifice necessarily requires slaughtering an animal from the category of anʿām (camels, cattle, and sheep/goats) within a specified time.
Rather, that is considered a form of charity (ṣadaqah) for which one is rewarded, but not an udhiyah. And Allah Almighty knows best.
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.