Does the intention (niyyah) of the principal (muwakkil) suffice for the sacrifice (dhabḥ) of the udhiyyah?
Praise be to Allah, and peace and blessings upon our master, the Messenger of Allah.
If a person appoints an agent (wakīl) to perform the slaughter (dhabḥ) on their behalf, the intention (niyyah) of the principal (the one appointing the agent) suffices, and there is no need for the agent to have this intention himself. In fact, even if the agent is unaware that he is performing the sacrifice on someone's behalf as an udhiyyah (sacrificial offering), this does not affect its validity. And Allah, the Most High, knows best.
What is the ruling on making up the regular pre-Dhuhr *sunnah* (*al-sunnah al-rātibah al-qabliyyah*) for one who missed it in the Dhuhr prayer?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah.
Every timed supererogatory prayer (nāfilah) is recommended (mandūb) to be made up if its time passes. As for a nāfilah that precedes the obligatory prayer, performing it before the farḍ is a matter of proper etiquette (adab); if it is prayed after the farḍ, it is still considered adāʾ (performed within its time). If the time of the farḍ has ended, it is recommended to make up (qaḍāʾ) the nāfilah for one who did not pray it.
Whoever misses the sunnah before Dhuhr, whether with an excuse or without one, it is recommended for him to pray it after the obligatory Dhuhr prayer, within the time of Dhuhr, even at its end, and it counts as adāʾ, not qaḍāʾ. It is also recommended for one who missed the pre-Dhuhr sunnah to pray it before the post-Dhuhr sunnah, in observance of the proper order. It is stated in ʿUmdat al-Sālik wa ʿUddat al-Nāsik (Vol.1/p.60): "What precedes the obligatory prayer has the same time as the obligatory prayer, and performing it before the farḍ is a matter of etiquette; after it, it is adāʾ. What follows the farḍ has its time begin with the performance of the farḍ and end with the end of its time."
If the time of Dhuhr has ended, it is recommended for him to pray both the pre- and post-Dhuhr sunnahs, and they count as qaḍāʾ for both. It is stated in ʿUmdat al-Sālik (1/61): "Every timed nafl, such as the ʿĪd prayer, Duha, Witr, and the regular rawātib of the obligatory prayers, if it is missed, it is recommended to make it up forever." And Allah Almighty knows best.
What is the ruling on congregational prayer performed outside the mosque, and is it a condition that it be performed in a mosque in order to attain the full reward?
All praise is due to Allah, and peace and blessings be upon our master, the Messenger of Allah ﷺ.
It is not a condition for attaining the reward of congregational prayer that it be performed in a mosque; however, a Muslim should still be keen to perform his prayers in the mosque, so as not to miss out on immense reward. This is due to the virtue of the mosque and of spending time there (i‘tikāf), the typically larger number of worshippers present, and the additional reward gained through walking to and from the mosque. On the authority of Abū Hurayrah (raḍiyallāhu ‘anh), the Messenger of Allah ﷺ said: "Whoever purifies himself in his house, then walks to one of the houses of Allah to fulfill one of the obligations Allah has prescribed, one of his steps will erase a sin, and the other will raise him a degree [in status]." (Related by Muslim)
Whoever misses congregational prayer at the mosque should still strive to pray in congregation at his workplace, or together with the members of his household. And Allah, the Exalted, knows best.
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.