Or $(\mathbb{N}\setminus\{1\},+)$, as some people prefer to call that monoid.

I don’t have time just now to write a full answer, but doesn’t this follow from Bergman’s theorem that every monoid is the monoid of finitely generated projective modules for some ring unless it obviously isn’t?

What's the definition of "smooth" in this context?

I don't see how the sentence you quote from Felix, Halperin and Thomas is claiming that the property of formality can be read from the cohomology algebra.

The sentence after Question 1 has a few typos, and I'm not quite sure that I understand what you meant to write. In particular, when you say "The additive group", do you mean "The automorphism group of the additive group"? So are you just pointing out that the additive group of a division algebra is a 2-orbit group? The reason that I think this might not be what you meant is that then it's not clear why you mentioned division algebras, as the additive group of any nonzero vector space is a 2-orbit group.

I hope they followed pure mathematical norms and had their names listed in alphabetical order.

Do you mean $2^{\lfloor{\rm log}_2 n\rfloor}$ rather than $\lfloor{\rm log}_2 n\rfloor$?

@BenjaminSteinberg Anyway, I'll edit my answer to make it clear that I wasn't assuming your definition.

@BenjaminSteinberg I didn't know that. I guess it makes sense, although personally I'd have chosen a different name than "simple", as it seems a bit confusing to use "simple module" for anything other than "simple object in the module category". I checked some of the books on my shelf, and your definition does seem quite common. But not universal. Notably, in Jacobson's Lectures in abstract algebra it is set as an exercise to prove that if $M$ is a simple left $R$-module then either $RM=M$ or $RM=0$ with $M$ cyclic of prime order. But I didn't find any very modern references.

Should a (left) semisimple nonunital ring be a finite direct sum of minimal left ideals? (This is automatic in the unital case.)

Henning Kause, in Chapter 14 of his book Homological Theory of Representations, covers the “small” version of all this (i.e., for Serre, rather than localizing, subcategories of small abelian categories) in some detail. I’m not sure if there are added difficulties in the “localizing” case.

@Mare Yes, it has dominant dimension zero.