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 a wife considered sinful if she asked for divorce because her husband married another woman?
A second marriage is lawful, and so it is impermissible for you to ask for divorce because of that. If you are certain that you won`t be able to deliver Allah`s rights and those of your husband`s , then it is permissible for you to ask for Khul`a (Divorce on the demand of the wife). However, you should exercise patience since after difficulties, there is easiness by the will of Allah.
A woman has asked her husband for Khulu`, her gold, deferred portion of the dowry and furniture. Is she entitled to that, knowing that her husband doesn`t want to divorce her?
All perfect praise be to Allah the Lord of the Worlds. May His peace and blessings be upon Prophet Mohammad and upon all his family and companions.
Khulu`* can`t takes place save before a judge of Sharia and he has the authority to make the suitable decision in this matter. And Allah The Almighty Knows Best.
* Khulu' is when parties agree to separate by way of consent, usually upon terms such as the wife agreeing to repay her Mahr (dowry) to the husband upon him agreeing to grant Talaq. The iddah period (waiting time after a divorce) of a woman who seeks a khula, is one menstrual cycle or one month if she is post-menopause i.e. ceased menstruating. This is to ensure she is not pregnant. This differs from when a man gives a talaq; the iddah period is three cycles or three months.
Is the Aqeeqa due on the newborn in case of dying almost a month after the delivery?
Offering Aqeeqa after a week of a newborn death is recommended.