What is the meaning of the Udhiyah?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The Udhiyah (Sacrificial Offering) refers to the livestock (An'am) that is slaughtered as an act of drawing closer to Allah the Almighty. This takes place on the day of Eid al-Adha (the 10th of Dhu al-Hijjah) and during the three days of Tashreeq that follow the day of Eid. And Allah the Exalted knows best.
Is it permissible for a woman to offer the obligatory prayer after Adhaan (call for prayer), and before Iqamah (The announcement that the prayer is about to begin) in the mosque?
Yes, it is permissible for a woman to offer the obligatory prayer immediately after the Adhan and before the Iqamah in the mosque. And Allah Knows Best.
What is the ruling on offering the sacrifice (uḍḥiyah) using the wealth of a minor or a legally incompetent person (safīh)?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
It is not permissible for the guardian (walī) of an orphan or a legally incompetent person (safīh) to offer the sacrifice (uḍḥiyah) on behalf of the minor or the safīh using their wealth, because the guardian is charged with exercising caution over their wealth and is prohibited from giving any of it away voluntarily, and the udḥiyah is a voluntary act of charity (tabarru'). The guardian may, however, offer the sacrifice using his own personal wealth. And Allah, the Most High, knows best.
What is the ruling on speaking while relieving oneself?
It is disliked for one relieving himself to speak while doing so, except out of necessity. And Allah the Almighty knows best.