Is permissible to buy/sell with non-Muslim?
Yes buying/selling for a Muslim with non-Muslim is permissible as regards what is lawful.
I`m a cashier at a branch of the Civil Service Consumer Corporation. Sometimes, customers pay more than the due amount. Do I have the right to take that extra money knowing that I cover shortages from my own salary?
All perfect praise be to Allah the Lord of the Worlds. May Allah`s peace and blessings be upon our Prophet Mohammad and upon all his family and companions.
If customers give that money willingly to help you with the potential loss then this is permissible. And Allah The Almighty Knows Best.
Is it permissible for a wife to give her money to her family as a charity, or a gift without asking her husband, or seeking his consent ?
The wife has the right to give her money as a charity, or a gift to her family, or to other people after consulting her husband out of respect, and this is the meaning of treating on footing of kindness and equity. Therefore, if he wanted to stop her from helping her family, then there is no harm in not telling him.
Is it permissible for a Muslim physician to specialize in gynecology obstetrics, and what is the ruling on working as such?
Some fields of knowledge are an individual duty while others are a collective one, and specializing in gynecology obstetrics is a collective duty. However, if there were female physicians to treat women, then there is no need for a male gynecology obstetrics specialist to look at women`s Awrahs (private parts) except in necessary situations since the jurisprudential maxim says:” Necessity must only be assessed and answered proportionately.” And Allah Knows Best.