Your cousin's half-brother may or may not be related to you. Your cousin has two parents, only one of whom is related to you (i.e. is the sibling of one of your parents). You cousin and his half-brother have only one parent in common. If the half-brother is the child of you parent's sibling, then the half-brother is related to you and is your cousin, too. If the half-brother is not the child of your parent's sibling, then he is not related to you at all.
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There is no relationship category of "half-cousin." There are half-brothers and half-sisters because the sibling relationship requires two parents in common. When only one parent is in common, their offspring are called "half-" The cousin relationship, on the other hand, requires only one connection. Your First Cousin is the child of one of your parent's siblings. So, if you have a Cousin who is the child of the sister of your father, and your father also had a brother, that brother is your cousin's uncle and is also your uncle. However, your cousin may also have an uncle who is the child of her father, rather than her mother. That uncle is not your uncle, and is not related to you at all. If your uncle has children by two different marriages, all those children are your cousins, regardless of who their mother is. If the wife in that second marriage also had children from a previous marriage, none of them are your cousins in the strictest sense. If you really like them and want to consider them family, of course there is nothing preventing you from calling them "cousins", too. But "half-cousin" has no meaning. If you want something other than "cousin" try "step cousin" as their mother would be Step-mother to your uncle's children from his first marriage.
Those of your half-brother's cousins who are his cousins through the parent you share are also your cousins and related to you. Your half-brother also has cousins through his parent who is not also your parent, and those cousins are not related to you.
No, because you share no ancestors. However some people like to say he would be your uncle-in-law.
There is no relationship in the English language that is called a "half cousin." Everyone who is your cousin is related to you through one and only one parent, so, unlike a "half brother" there can be no such thing as a "half cousin." If you have a first cousin (child of your aunt or uncle), then your child and that first cousin are first cousins once removed. If you have a second cousin, they your child and that second cousin are second cousins once removed.
Your first cousin's grandchild is your first cousin, twice removed.
Lets say you only have ONE first cousin... Your mom and your cousin's mom are sisters. Your father and your cousin's father are brothers. Doubling the genes you have in common... or your mom and your cousin's dad are brother and sister... and your dad and your cousin's mom are brother and sister...
The grandson of your first cousin and your grandson are third cousins to eachother.
Your father's brother's child is your cousin. A first cousin is anyone who shares one or more common grandparents with you.