It may be a problem that the existence of a length function with the properties you want is rare and most monoids do not possess such a thing.

A version of the Nullstellensatz that meets the requirements says "Every field that is finitely generated as a ring is finite". This is also used in Luc Guyot's argument and he gives a reference. This can be applied to conclude that $J/I$ is finite because there is a maximal ideal $M$ of $A$ such that $A/M$ is isomorphic to $J/I$. The Artin Rees Lemma says that the $M$-adic topology on $A/I$ induced the $M$-adic topology on $J/I$ and hence there is a natural number $n$ such that $M^n$ annihilates $A/I$. This shows that $A/I$ is both finite and local.

@JoelSpringer It's not clear what $H^n(U)$ means in the edit: note that singular cohomology satisfies the Eilenberg--Steenrod axioms so if $H^n(U)$ denotes singular cohomology of $U$ then you get singular cohomology, instead of an alternative approach to Čech cohomology.

When passing to the dual $A^*:=\hom_k(A,k)$ you switch from left modules to right modules and right modules to left modules. So you maybe want to use the double dual. If $M$ is irreducible then so is $M^*$. Thus there is a surjection $A\twoheadrightarrow M^*$ and thus there is an embedding $M=M^{**}\rightarrowtail A^*$.

You mention the definition of Čech cohomology in terms of nerves of coverings. What is the second definition of Čech cohomology that you want to compare this with?

Up to isomorphism there are continuously many (i.e. $2^{\aleph_0}$) countably infinite groups.

I think it would be useful to make explicit what is the problem with the Kan extension in the particular case when $X=\{s,\eta\}$ is the $2$-point non-$T_1$ space in which only the point $s$ is closed. I don't yet see why the Kan extension disagrees with $\underline{X}$ in this particular case.

Where, in this argument, is the T1 separation argument used?