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.
Does tooth extraction during the day in Ramadan break the fast?
Simply extracting a tooth during the day in Ramadan does not break the fast. However, if water or blood enters the body cavity, the fast becomes invalid.
Whoever's fast is invalidated in this way must refrain from eating and drinking for the rest of the day out of respect for the sacred month and make up for that day later.
It is preferable to postpone the extraction until nighttime or after Ramadan if possible.
Is it permissible for a woman to observe voluntary fast without her husband's permission?
A woman is not permitted to observe a voluntary fast (nafl) while her husband is present without his permission.
Can I start fasting based on the moon sighting of another country?
You must follow the moon sighting of the country you are in.