Is the one offering the sacrifice liable if its meat spoils?
Praise be to Allah, and prayers and peace be upon our Master the Messenger of Allah.
If the meat (of the sacrifice) spoils due to the negligence of the one offering the sacrifice in preserving it, or due to improper storage, then he is liable to compensate for the portion due to the poor (which is estimated as half a kilogram of meat). If the sacrifice was a vowed (mandatory) one, then he is liable for all of it.
However, if he was not negligent, then there is no liability upon him, because its ruling is the ruling of a trust (like an item left in someone's care). And Allah Almighty knows best.
What is the waiting period ('Iddah) for a woman whose husband has passed away, and what is the ruling on her wearing gold?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
The waiting period ('Iddah) for a woman whose husband has passed away is four months and ten days for one who is not pregnant. As for a pregnant woman, her waiting period lasts until she gives birth. It is obligatory for her to remain in the marital home, only leaving for a necessity. During this time, it is prohibited (Haram) to display any form of adornment on the body or clothing; this includes wearing kohl, gold, all types of perfume, and dyeing the hair. Likewise, it is prohibited to receive a direct marriage proposal or to marry during this period.
It was narrated by Umm 'Atiyyah that the Prophet ﷺ said: 'We were forbidden to mourn for a deceased person for more than three days, except for a husband, for whom the mourning period is four months and ten days. During this time, we were not to use kohl, nor wear perfume, nor wear dyed clothing except for garments made of 'Asb (coarsely dyed yarn). We were, however, granted a concession at the time of purification—when one of us bathed following her menses—to use a small amount of Kust (costus) or Azfar (fragrant substances). We were also forbidden from following funeral processions.' (Narrated by Al-Bukhari). And Allah the Exalted knows best."
What is the ruling on performing Tarawih in sets of four rak‘ahs with only one Tasleem?
It is not permissible to combine more than two rak‘ahs with a single Tasleem in Tarawih prayer. Whoever does so, their prayer is invalid. Shaykh al-Islam Imam Muhammad al-Ramli (may Allah have mercy on him) stated: "If a person prays four rak‘ahs of the Tarawih prayer with a single Tasleem, it is not valid if they did so intentionally and with knowledge [of the prohibition]. Otherwise, it becomes an 'absolute voluntary prayer' (nafl mutlaq); because it is contrary to what has been prescribed." [Nihayat al-Muhtaj, Vol.2/P.127].
What are the categories of the livestock permissible for Udhiyah?
In the Name of Allah, and may peace and blessings be upon our Master, the Messenger of Allah.
An Udhiyah (sacrificial offering) is only valid if it is selected from Al-An'am (livestock), which includes: camels, cattle, and sheep or goats—whether male or female, and even if castrated. Allah Almighty says {what means}: "And for all religion We have appointed a rite [of sacrifice] that they may mention the name of Allah over what He has provided for them of [sacrificial] livestock." (Al-Hajj/ 34)
In terms of preference, camels are considered the best choice, followed by cattle, then sheep, and finally goats.
A single sheep or goat is valid as a sacrifice on behalf of one person. However, a single camel or cow can be shared by up to seven people. This is based on the narration of Jabir (may Allah be pleased with him), who said: "In the year of Al-Hudaybiyah, we sacrificed along with the Messenger of Allah (peace and blessings be upon him) a camel on behalf of seven people, and a cow on behalf of seven people." (Narrated by Muslim)
(Note: The term "Badanah" used in the original Arabic text specifically refers to a camel in this context). And Allah the Almighty Knows Best.