What type of illness that permits breaking the fast in Ramadan?
It is an illness that, if one fasts, there is a fear it may lead to their death or cause unbearable hardship.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.
Does smoking break the fast?
Yes, smoking breaks the fast because smoke particles intentionally enter the lungs, which are considered part of the body cavity (jauf).
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.