Some people say that a half uncle is your step-dad or step-mum's brother. Since, an uncle is the brother of your parent, the brother of a setp-parent would be a step-uncle.
In fact, however, neither "Half Uncle" nor "Step Uncle" are frequently or widely used in the English language. The sibling of a step parent is generally not considered to be a relation to you.
Your half uncle is the sibling of one of your parents, but not both. This means you share one biological parent with your half uncle.
Your mother's half-brother is your uncle, or half-uncle if you want to be exceedingly precise.
NO
YES......If your Mother of Father has Half-Siblings (Half Brothers or Half Sisters) and those siblings, (your Half Aunt(s) or Half Uncle(s) ) have children, those children become your Half Cousin(s). Saying that someone is your 'Half Aunt' or 'Half Uncle' or 'Half Cousin' is the correct term, but it is not put into practice very often and people just refer to their 'Half Aunt' or 'Half Uncle' or 'Half Cousin' as simply; 'Aunt' or 'Uncle' or 'Cousin'
YES......If your Mother of Father has Half-Siblings (Half Brothers or Half Sisters) and those siblings, (your Half Aunt(s) or Half Uncle(s) ) have children, those children become your Half Cousin(s). Saying that someone is your 'Half Aunt' or 'Half Uncle' or 'Half Cousin' is the correct term, but it is not put into practice very often and people just refer to their 'Half Aunt' or 'Half Uncle' or 'Half Cousin' as simply; 'Aunt' or 'Uncle' or 'Cousin'
Since the cousin is from your half uncle's wife's side of the family the cousin would not be your cousin.
A half-niece and her uncle are expected to share about 12.5 of their DNA.
A person and their half-uncle are expected to share about 12.5 of their DNA.
your uncle or even half uncle but it would be no different.
Uncle
Your grandfather's half brother is you great uncle. Some would insist on calling him your half-great uncle, but the English language has not had a need to get that specific and complex about such distant relationships.
A Great half uncle.