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.
Should water be put in the deceased`s mouth and nose while washing him/her?
Putting water in the mouth and nose of the deceased while washing him is impermissible.
Does the clipping of nails nullify ablution?
No, it does not, and it is preferable to wash the hands afterwards.
A sixteen-year-old young woman suffers from Alopecia, and lost her hair and eyebrows as a result. Is it allowed to draw her eyebrows using henna and an eyeliner?
All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
It is permissible for women to use henna and what the asker has mentioned isn`t considered hair extension. However, if this young woman is proposed to, she must disclose this matter to the person proposing to her to avoid deception. And Allah The Almighty Knows Best.