Can a person required to give kaffarah feed it to their own family members?
The kaffarah must be given to the poor and needy who are not financially dependent on the one giving the kaffarah.
If a person feeds it to their own family members, it does not count as kaffarah, and their obligation remains unfulfilled.
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 to pay the Fitr Zakah (obligatory charity) of Ramadan on behalf of a dead person?
The Fitr Zakah of Ramadhaan isn`t due on one who had passed away before the sunset of the last day of Ramadan. And Allah Knows Best.
Is it permissible to offer mandatory prayer while sitting?
All perfect praise be to Allah, The Lord of The Worlds, and may His Peace and Blessings be upon our Prophet Muhammad and upon all of his family and companions. Standing up is a pillar in the mandatory prayer and the prayer is not valid without it unless one cannot do it. As for Nafila (supererogatory prayer), one can perform it in the state of sitting even if he/she can stand, but his/her reward is half the reward of the person who prays while standing if he/sh has no legal excuse for such an act. And Allah Knows Best.