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 required of a traveler or a sick person if they broke their fast and then their excuse ceased?
If a traveler settles or a sick person recovers after having broken their fast, it is recommended for them to refrain from eating and drinking for the rest of the day, but it is not obligatory.
Do ear and nose drops break the fast?
Yes, ear and nose drops break the fast because both the nose and ears are open passages to the body cavity.
What is the ruling on swallowing phlegm while fasting?
If a fasting person intentionally swallows phlegm, their fast is invalid, as it is possible to avoid it.
However, if they swallow it unintentionally, their fast remains valid.
Phlegm is impure and should be avoided.