Yes. The grandparents of one are the great-grandparents of the other.
Yes. They are your First Cousins, twice removed.
Your Father's first cousins are also your first cousins, once removed.
They are out of synch by a generation. They would be first cousins once removed.
You are first cousins once removed to each other.
You and your first cousin's children are first cousins once removed. Your children are their second cousins.
The nephews of your first cousin once removed, like the children of your first cousin once removed, are your first cousins twice removed.
First, some of your cousins are not related to your grandmother at all. That is, your father's mother is not related to the cousins you have who are the children of your mother'ssiblings.On the same side of the family, your first cousins are your grandmother's other grandchildren. Your second cousins are your grandmother's great nieces and great nephews. Your third cousins are your grandmother's first cousins, twice removed, your fourth cousins are your grandmothers second cousins, twice removed, etc., etc.
Yes, being cousins means that you are related.
Your son and your niece are first cousins. Her daughter and your son are first cousins, once removed.
They are out of synch to a common ancestor by one generation. That would make the first cousins once removed.
First cousins three times removed.
The English language has no special term for the cousin of a cousin. Either that person is related to you, and so is describable by that direct relationship as some sort of cousin (or a sibling), or is not related to you.