"isomorphic" in what sense? As sets, you can even take $n=1$ and $\sim$ to be trivial; this is indeed just a statement of countability. If you mean as a group, then this is not possible, since $\mathbb A$ is infinite-dimensional over $\mathbb Q$.

There have been some successful learning roadmap questions before, I think formulating yours in such a way may be more productive. The road to get there may not be particularly straightforward.

Chen's theorem is a fairly advanced result in sieve theory, I think even by modern standards. You are much better off studying from textbooks first, and looking into some simpler research applications then. I am not in position to give you specific references, but I heard good things about Opera de Cribro.

It's definitely an interesting proposal and I'm sure some people have thought about it. To my knowledge, there is no remotely successful approach of this form, not even conjecturally.

Penelope Maddy, Believing the Axioms. II

This moduli space is not a variety or a scheme, which you can see for instance since the map from $K$-points to $L$-points is not an injection for $L/K$ a field extension. What type of answer do you hope for?

You may need to restrict to some nice category like that of CW complexes. I suspect a characterization analogous to Eilenberg-Steenrod axioms may exist but I have never encountered one. Few properties you probably want to include is homotopy invariance, Seifert-van Kampen and evaluation at $S^1$. Whether this is sufficient, I have no idea.

"Cayley graph" of a group is not well-defined, it depends on the choice of a generating set of the group, and different choices may give graphs with different automorphism groups.

Isn't a bi-infinite path an example?

Using the structure over $\mathbb Q(a)$ a bit more we can also construct a basis as at the start of your question. I've written that up as an answer to your MSE question.

"approximately equal" is not very well-defined. What exactly are you asking?