Is it obligatory to have an intention (niyyah) for every prayer?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is obligatory for the worshiper to have a specific intention (Niyyah) for every prayer, meaning they must consciously intend the act of worship they are performing. The intention is a pillar (Rukn) of the prayer, and the prayer is not valid without it. It is not a requirement to utter the intention verbally; rather, doing so is considered a recommended Sunnah. And Allah the Exalted knows best.
What are the categories of the livestock permissible for Udhiyah?
Praise be to Allah, and peace and blessings be upon the Messenger of Allah.
The Udhiyah (sacrificial offering) is not valid except from Al-An’am (livestock), which are: camels, cattle, and sheep/goats. Allah the Almighty says {what means}: "And for every nation We have appointed a rite [of sacrifice] that they may mention the name of Allah over what He has provided for them of beast-livestock." [Al-Hajj/34]
The best of them are camels, then cattle, then sheep/goats. A sacrifice of a camel or a cow avails for seven people. Jabir (may Allah be pleased with him) said: "We sacrificed with the Messenger of Allah (peace and blessings be upon him) in the year of Al-Hudaybiyah a camel for seven and a cow for seven." [Narrated by Muslim] And Allah the Almighty knows best.
Is it permissible for a praying person to make supplication in each Sajdah (a prostration) of obligatory prayer?
It is permissible for a praying person, whether praying individually, or in congregation, to make supplication during Sujood (prostration) in obligatory, or voluntary prayers. However, it is disliked for the Imam to do so since he is required to spare the prayer performers any sort of hardship, unless he is leading a certain group who don`t mind him extending the prayer.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.