When does the time for Udhiyah begin?
Praise be to Allah, and peace and blessings be upon our master, the Messenger of Allah.
The time for Udhiyah (sacrificial offering) begins once the sun has risen on the day of Eid al-Adha—which is the tenth of Dhul-Hijjah—and a period of time has passed equivalent to two brief prayer units (Raka'at) and two brief sermons (Khutbah). It then continues until the sunset of the last of the days of Tashreeq, which are the eleventh, twelfth, and thirteenth of Dhul-Hijjah.
The Prophet (peace be upon him) said: "All the mountain passes of Mina are places of sacrifice, and in all the days of Tashreeq there is slaughtering." [Narrated by Al-Bayhaqi and Ibn Hibban].
The best time to slaughter is after finishing the Eid prayer, due to the saying of the Prophet (peace be upon him):
"Indeed, the first thing we begin with on this day of ours is to pray, then we return and slaughter. Whoever does that has attained our Sunnah, and whoever slaughters before [the prayer], it is only meat he has presented to his family; it is not part of the ritual sacrifice (Nusuk) in any way." [ٌReported by Bukhari & Muslim].
It is valid to sacrifice at any time, whether by night or by day; however, it is disliked (Makruh) at night. And Allah the Almighty knows best.
Is it permissible for a wife to leave her house without asking her husband?
It is impermissible for the wife to leave her house without asking her husband except for a sound reason.
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. And Allah Almighty knows best.
Is vomit among the nullifiers of Wudu (ablution)?
Vomit does not nullify Wudu, but it is a Najaasah (impurity) that requires rinsing the mouth and washing whatever became dirty by it since the prayer becomes valid only after the removal of Najaasah. And Allah Knows Best.