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.
Is the prayer of someone who didn`t face Quibla (Ka`ba direction) valid?
All perfect praise be to Allah,The Lord of The Worlds. Facing the Quibla is a condition for the validity of prayer, and the prayer of the person who hadn`t faced the direction of the Quibla is invalid, and he is obliged to repeat it. And Allah Knows Best.
Does an internal medical examination for a woman affect her fast?
An internal medical examination for a woman invalidates the fast because it involves the entry of a foreign object into the body cavity (jauf) while fasting.
In this case, the woman must refrain from eating and drinking for the rest of the day out of respect for Ramadan and make up for the missed fast after Ramadan.
Such an examination should be avoided in Ramadan and other months unless absolutely necessary, as exposing the private parts is only permissible in cases of necessity.
If necessary, a woman should seek a Muslim female doctor first. If one is unavailable, she may see a female doctor from the People of the Book (Jews and Christians). If neither is available, she may consult a trustworthy and competent Muslim male doctor.
What is the ruling on using eye drops while fasting?
Eye drops do not break the fast, even if the taste reaches the throat, because the eye is not an open passage to the body cavity.