She is his sister-in-law.
Your mother-in-law's brother is your spouse's uncle, but is not related to you.
Yes.
You are not related to the wife your mother's sister's husband's brother.
They have the same mother and different fathers.
Your mother's husband is your father. His father-in-law is your mother's father, your grandfather. Your grandfather's son is your mother's brother, and your uncle. Your uncle's daughter is your first cousin.
Your brother's mother-in-law is not related to you, unless you also married one of her children.
Your mother's brother is your uncle and his wife is your aunt.
If your brother-in-law is the brother of your wife then his mother is your mother-in-law. If your brother-in-law is the husband of your sister, his mother is not related to you but is your sister's mother-in-law.
You are not related at all.
The sister-in-law of your father's only brother could be your mother. If, instead, she is the sister of your uncle's wife, she is not related to you at all. You share no common ancestor with her.
No.
If you have different fathers, you may be related if you share some other ancestor, such as a mother, a grandparent, great grandparent, etc.