Is it permissible to offer prayer at home, or should it be offered in the mosque?
All perfect praise be to Allah, The Lord of The Worlds. It isn`t preferable for the man to pray at home as praying in the mosque is twenty seven times more rewarding. Therefore, this should motivate him to offer prayers in the mosque. And Allah Knows Best.
Is it required for a discerning child to make the intention at night if they wish to fast?
If a discerning child wishes to fast during Ramadan, they must make the intention at night because intention is one of the pillars of fasting, even though fasting is not obligatory for them.
What is recommended for a Muslim to say on Laylat al-Qadr?
On Laylat al-Qadr, it is recommended for a Muslim to recite the supplication (du‘a) that the Prophet ﷺ taught Aisha (may Allah be pleased with her):
"O Allah, You are Most Forgiving, and You love forgiveness, so forgive me." [Narrated by Al-Tirmidhi]
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.