Is it incumbent on a family to provide for its old handicapped son and take care of him?
The family and relatives - e.g. brothers - of such a person are obliged to spend on him and take care of him if he didn`t have money of his own.
What is the ruling of Sharia on uncovering Awrah before a female laser-hair-removal specialist to remove thick hair knowing that I`am engaged and feel embarrassed and this treatment is guaranteed and definitive?
Praise be to Allah the Lord of the Worlds. May His peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
It is lawful for a Muslim woman to look at the body of another Muslim woman except what is between the navel and the knees. In addition, it is lawful to look at the whole body for treatment; however, having thick hair isn`t considered an illness. And Allah The Almighty Knows Best.
Is it permissible for a woman to shake hands with her brother-in-law?
It is forbidden for the woman to shake hands with her brother-in-law because he isn`t a Mahram (unmarriageable) of hers.
Is it permissible for the young woman to turn the suitor down if she had no feelings for him?
Yes, it is permissible for her to do so.