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 for one to shake hands with his uncle`s daughter?
No, it isn`t because he is a non-Mahram (Marriageable) to her.
What is the ruling on public street water if it gets on a person's clothes or body?
The default ruling is that public street water is pure (tahir). If one is certain of its impurity (najasa), then a small amount that gets on the lower part of a person's clothes or body is overlooked (excused). And Allah the Almighty knows best.
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.