I’ll assume half-cousins count; my dad has a half-sister with two kids which I guess makes them my half-cousins.
My mom’s brother has an adopted daughter and I am going to count her, even though we are not blood related.
My dad’s brother was married several times, so I’m just going to have to estimate the number of offspring he has. 
I still think that only gets me to six, total.
None - both parents were only children. (Each of my nieces and nephews, on the other hand, have five of them.)
Speaking of first cousins - well, first cousins once removed, anyway - is there a word to distinguish the first cousin once removed that is the grandchild of the common ancestor as opposed to the great-grandchild?
I’m not sure if I understand the question, but a first cousin shares a grandparent. If you share a great-grandparent but not a grandparent, you are second cousins. If one of you has a grandparent who is the great-grandparent of the other, you are first cousins once removed. (So the child of your first cousin is your first cousin once removed.)
I always thought first cousins share common grandparents (plural), never occurred to me consider what you’re saying.
So: my mother’s half-siblings have several children, named Jane, Joan, John, and Jimmy.*
JJJ&J and I have one common grandparent- my mother’s father. Are JJJ&J and TC first cousins?
I have 43.
None are deceased.
5 maternal + 4 paternal = 9.
Yes, they are first cousins. You could call them half first cousins if you want, but that just seems unnecessary to me (maybe some people would find it important).
I support that nomenclature, too. The cousin I mentioned upthread is the daughter of my blood aunt who married this cousin’s father after she divorced the father of the older two kids. So the blood rrelationship to me is the same (isn’t it?) as if she had mothered all three from the same father. From the kids’ point of view the older two are less kin to the third child than they are to each other since they share the father as well as the mother. Half-siblings, for sure. But my cousin relation to the third child remains (as best I can tell) unaffected.
Now if the older two siblings’ father (not blood kin to me) had had another child after that divorce, it would still be a half-sibling to my first cousins, but the best I could call it (the new kid) would be half-cousin and I wouldn’t count it as a blood relative at all. Maybe legal relative or “in-law?” It gets fuzzy to me. 
Right, but that is not the same scenario. The question was whether an aunt or uncle who is a half-sister/half-brother to your parent has kids who are your first cousins or something else. Those cousins would only share one grandparent, not two.
In the case you’re talking about, if your aunt or uncle who is a full sister/brother to your parent has kids with two different people, all of the kids would bear the same relationship to you.
Two. One five years older than me and one about 25 years younger than me. They’re half-siblings.
My Mom had four sisters asnd three brothers; only one brother had no kids. My Dad had one sister, who had only one son.
Oddly enough, my answer has to be “don’t know” – for which there isn’t a voting slot. I have six first cousins on my mother’s side, all still alive. For complicated reasons, I don’t know much about my father’s side of the family, and have long been completely out of touch with them. (My parents are long dead, so are not there to ask.)
I know that I have, or have had, some first cousins on my father’s side; he had several siblings, some at least of whom married and reproduced. But there’s only one such, who I can put a name to; and I’ve no idea how many others there may be / have been.
Crap, I lied. I knew there was something wiggling at the back of my brain. My one uncle did have a kid when he was very young, and then broke up with the mom. I never knew the cousin (I think it was a boy) but he counts. It moves me from one bracket to another in the poll.