Right, this is true on the nose in the cancellative case only. In the non-cancellative case this is true in a derived sense, if you define the localization in an infinity-categorical sense. (See e.g. ncatlab.org/nlab/show/group+completion or the Dwyer-Kan paper referenced there.)

I see. That will actually always be equivalent to the classifying space for the group completion, essentially since the classifying space construction doesn't distinguish "forwards" and "backwards" arrows

The cohomology Ext_M(Z,A) most certainly is computable by pullback to the group completion! The module Z is supported in the open Z[M^{gp}], and so any Ext can be computed after localizing to this open. By "same as the categorical cohomology" I mean it is the same as the cohomology of the trivial representation of the corresponding category (not sure if this is what you mean by "classifying space as a category")

Maybe more generally, a very good first thing to do when working with a commutative monoid M is to pass to the corresponding "toric" geometry object X = Spec Z[M]. The category of M-representations is equivalent to the category of quasicoherent sheaves on X, and invariants computed in this category have geometric meaning. If you want to encode the Hopf algebra of M and not just its underlying ring, this is equivalent to considering its spectrum X as a semigroup object. Most interesting monoid invariants should have meaning as invariants of commutative geometric semigroups.

What is the use case that you have in mind? The most straightforward generalization of group cohomology would be Ext_M(Z, Z) which is the same as the categorical cohomology. The problem is that this is a boring invariant in the commutative cancellative case since it is invariant under group completion. Indeed, transposing to geometry, this is the self-Ext of the skyscraper sheaf at 1 in Spec(Z[M]), which can be computed inside the open chart Spec(Z[M^gp]) corresponding to the group completion.

I do not recommend using the term "symmetric/anti-symmetric tensor product", which is confusing and non-standard, since there is no such thing as a symmetric or anti-symmetric tensor product of two different representations. Write "symmetric or exterior power" instead.

Unless I am misunderstanding your question, you can just choose a sequence of values $x_n$ and define $f(x)$ to be the $n$th partial sum in your expression between $x_n$ and $x_{n+1}$. As long as $x_n$ go to zero sufficiently quickly (for example $x_n = 10^{-n^2}$) the function will have limit 1 at $x=0$ and satisfy your little-o condition

@DavidBen-Zvi Good point! Yes, I think you're right: I think you can for example say that an O-algebra in C is an algebra of O in the (category, object) pair category over a given O-monoidal C

Thank you! That certainly seems to settle it. Interestingly enough, I had recently read the first couple of sections of the Chu-Haugseng paper, but didn't notice this theorem.