Simply called: Ex-brother in law
Your wife's brother is your brother-in-law. His son's daughter is his granddaughter and your child's first cousin, once removed, but is not related to you. You can refer to her as "my brother-in-law's granddaughter."
Yes, you are related to your brother-in-law's grandson. Your brother-in-law's grandson would be your nephew once removed. This means that you are not directly related by blood, but you share a family connection through your brother-in-law's family.
Your nephew's fatehr is either your brother or your brother-in-law.
The English language has not special name for the cousins of your spouse. You could just call them by their first names like you would your friends. If you really like them, you could decide to call them your cousins.
Your brother-in-law's first cousin's daughter is hisfirst cousin, once removed. However, the English language has no term or phrase describing your relationship to that person, with whom you share no ancestry.
A boy and his mother's first cousin once removed are second cousins or first cousins twice removed. Which it is depends on whether the mom's first cousin once removed is the child of her first cousin or the mother of her second cousin. My first cousin, Mary, has a son, David, who is my 1st cousin, once removed. He is my daughter's 2nd cousin.
Your cousin's brother is also your cousin. His child is your cousin once removed. For example, if your first cousin has a child, that child is your first cousin, once removed. If your third cousin has a child, the child is your third cousin, once removed.
Your grandfather's brother is your great uncle. His daughter (or his son) is your parent's first cousin and your first cousin, once removed.
Your grandfather's brother's daughter is your first cousin, once removed.
Yes
You aren't. Your mother's brother is your Uncle. Your Uncle's aunt is one of your grandparent's sisters - your grand Aunt. Her "only" niece could be your Uncle's sister or cousin, you don't provide enough information to know. If it's your Uncle's sister she is your Aunt. If it's your Uncle's cousin she is your first cousin, once removed. If it's your Aunt, her mother-in-law is simply the mother of the man she married. Your uncle-in-law's mom. The "in-law" thing does not carry past the Uncle-in-law, or you'd be "in-law" to the world. So no relation to his mom, or the mom's husband. If your cousin (once removed), her mother-in-law is simply the mother of the man she married. Your first cousin/once removed-in-law (not a generally recognized relation). Were it to be, it still doesn't go past the cousin/once removed-in-law, or you'd be "in-law" to the world. So no relation. Therefore no relation to his mom, or the mom's husband. (The phrasing "mother-in-law's husband" connotes that the man is a second husband and no relation to the Uncle-in-law or cousin/once removed-in-law - else such would have been described as "father-in-law" not "mother-in-law's husband. Therefore, even moreso NOT a relation.)
Your children are the first cousins of your brother's children, and the first cousins once removed of your brother's grandchildren.