Yes, "second cousin" refers to the grandchildren of your great aunt or great uncle.
If your second cousin's mother is related to you, she is your first cousin, once removed. Remember that one of your second cousin's parents will not be related to you.
You and your second cousin's child are second cousins once removed to each other.
Your first cousin's child would be your second cousin
All of your first cousin's children are your first cousins, once removed. All of your second cousin's children are your second cousins, once removed. It does not matter if the child is the first , second or tenth child. The English language has only the one term.
If your aunt is the sister of one of your parents, then her first cousin is your second cousin, her second cousin in your third cousin, and so forth. If you aunt is the wife of the brother of one of your parents, then her cousin is not related to you and the English language has no term for that person other than "my aunt's cousin."
According to Collins Dictionary, cousin : uncle :: second cousin : second uncle. The father of your cousin is your uncle, so the father of your second cousin is your second uncle. It is a way to remove some of the ambiguity of the "first cousin once removed" relationship. This term is used all over the world and is slowly gaining acceptance in the US due to its ability to distinguish the generation of each person.
In the sense of what is the name of the relationship, that would be "second cousin-in-law", though if you are that close to your second cousin and his wife, you could just say, "cousin". In the sense of how you would address her, it would be the same as with your second cousin. Just the name that she goes by. After all, "second cousin", or even "cousin", isn't a title like Uncle or Aunt is.
If you are his second cousin, he is your second cousin
Asking about your second cousin's cousin's cousin is asking about a complex but poorly defined relationship. Are the cousins of the second cousin his first cousin, second cousin, fifth cousin twice removed, or something else. It is likely that a cousin's cousin's cousin is not related to you at all. It is also possible that your second cousin's first cousin's first cousin is also your second cousin.
Your second cousin's niece is the daughter of your second cousin's brother or sister. Your second cousin's brother or sister is also your second cousin, so this question is the same as "How are you related to the daughter of your second cousin."The daughter of your second cousin is your second cousin, once removed.
Your cousin's daughter is called your first cousin, once removed. Although the term "second cousin" is sometimes used for this relationship, it is technically incorrect.
Your first cousin's second cousin may be your second cousin as well, or may not be related to you at all.