Well, I was focusing on OP's question about existence of limits and colimits, and the question of whether any model is complete. As Andrej said, "Because ZFC proves that all small diagrams have limits", and this was my answer too. I think we are all in agreement here.

Sorry I don't know a name for precisely what you want. It seems like a natural-enough notion.

I guess you mean $Fc$, not $Fd$. What example do you have in mind?

Harry, I don't know the answer to this question, but I'm inclined to agree with your intuition. There is a certain amount of work on which monads come from operads; in the case where the base category is $Set$, see for example appendix C.2 in Tom Leinster's book, where operads are described as equivalent to strongly regular finitary algebraic theories. There is also relevant information in the Appendix of Joyal's article in SLNM 1234; I don't recall whether his arguments handle a base category like vector spaces over Z mod 2, but if I were investigating this, these are places where I'd start.

I understand the SB theorem as saying that in the category of sets, if there exists monos from X to Y and from Y to X, there there exists an isomorphism between them. I understand the KT theorem as saying that if $f: X \to X$ is a monotone function on a sup-lattice, then $f$ has a fixed point. Is that what you mean as well?

My bad, and thanks for catching that Harry. The responsible thing would be to cross out the bogus proof and insert the correction, but I can't seem to figure out how to cross out. Anyway, I'll fix this.

I have updated my answer to point you to a slightly different proof.

That's not an algebra homomorphism -- it doesn't preserve the identity.

I wrote it that way in remembrance of the canonical bijection between linear maps $M \to \hom_F(V, V)$ and linear maps $M \otimes_F V \to V$, and (by specializing) between algebra maps and module structures. I was trying to segue from a question about algebras to a module-theoretic question.