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.
Is it permissible for the person who hasn`t made wudu` (state of minor ritual impurity) to perform the Adhan?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is disliked (Makruh) for a person in a state of minor ritual impurity (Hadath Asghar) to perform the Adhan. However, if he does so, his Adhan is considered valid and fulfills the sunnah of the Adhan despite it being disliked. And Allah the Exalted knows best.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
What is the ruling on giving the expiation of an oath (Kaffarat al-Yamin) to a charity that feeds the poor?
Praise be to Allah, and peace and blessings be upon our Master, the Messenger of Allah.
It is allowed for a Muslim to appoint a proxy (agent) to distribute the expiation (Kaffarah) on their behalf. Our jurists have explicitly stated the permissibility of delegation (Tawkil) in the distribution of Zakat, Kaffarah, and vows (Nadr).
It is stated in Mughni al-Muhtaj (Vol.3/P.237): 'One has the right to pay the Zakat of their wealth personally... and they also have the right to delegate it.' And Allah the Exalted knows best.