When does the time for Udhiyah begin?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
The permissible timeframe for Udhiyah (sacrificial offering) begins on the day of Eid al-Adha—the 10th of Dhul-Hijjah—once the sun has risen and a period of time sufficient to perform two brief prayer units (Rak'ahs) and two short sermons (Khutbahs) has passed. This window remains open until the sun sets on the final day of Tashreeq, which is the 13th of Dhul-Hijjah.
Our Master, the Messenger of Allah (peace and blessings be upon him), said: "Every valley of Mina is a place of sacrifice, and slaughtering may be done throughout all the days of Tashreeq." (Narrated by Al-Bayhaqi and Ibn Hibban)
The days of Tashreeq refer to the 11th, 12th, and 13th of Dhul-Hijjah.
The most virtuous time to perform the sacrifice is immediately after concluding the Eid prayer, based on the statement of the Prophet (peace and blessings be upon him): "The first thing we do on this day of ours is to pray, then we return and offer our sacrifice. Whoever does that has acted in accordance with our Sunnah (tradition), and whoever slaughters before that, it is merely meat he has provided for his family; it has nothing to do with the ritual sacrifice." (Reported by Al-Bukhari and Muslim)
What is meant here is an estimation of time rather than the actual performance of the prayer itself, as our Master, the Messenger of Allah (peace and blessings be upon him), used to offer the Eid al-Adha prayer immediately after sunrise.
The sacrifice is valid if performed at any time during these designated days, whether by day or by night, though slaughtering at night is considered disliked (Makruh). And Allah the Almighty Knows Best.
When is it Sunnah to slaughter the 'aqīqah?
All praise is due to Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
It is Sunnah for the 'aqīqah to be slaughtered on the seventh day from the birth of the newborn. According to the sounder position, the day of birth itself is counted as the first of the seven days. Thus, for example, if the child is born on a Saturday, the 'aqīqah is to be slaughtered on the following Friday. If the child is born at night, the count begins from the day that follows. And Allah Almighty knows best.
Is the father a Mahram (Non-marriageable) to his son`s mother-in-law?
The father is a non-Mahram (Marriageable) to his son`s mother-in-law, so it is impermissible for them to look at each other, or to have a seclusion (Khalwah).
Can a woman leading other women in Tarawih prayer recite aloud?
Yes, a woman leading other women in Tarawih prayer is allowed to recite aloud, as long as no non-mahram (other than unmarriageable kin) men can hear her.