Is Zakah due on a woman`s deposited gold?
Zakah (obligatory charity) is due on deposited gold if it reached Nissab (minimum amount liable for Zakah), and a lunar year had lapsed over owning it. The due Zakah is (2.5%). And Allah Knows Best.
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.
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.
Is it permissible for the man to go out wearing shorts?
All Perfect Praise is due to Allah, The Lord of The Worlds
It is permissible so long as his Awrah (private part), which is between his navel and knee, is covered. However, the regular wear traditions and modesty are to be observed. And Allah Knows Best