I may be not understanding this correctly, but I think this is impossible. Take $x \in A^{\otimes \mathbb{N}}$, and some $a \in A^{\otimes k}$ such that $x \approx_{\varepsilon} a$. Let $\sigma$ be some finite permutation such that $\sigma(\{1, \dots, k\}) \cap \{1, \dots, k\} = \emptyset$, and suppose that $\sigma(x) = x$. Then $\sigma(a) \approx_{\varepsilon} a$, but this just only holds if $a \approx 1 \otimes \dots \otimes 1$. Problem is $A^{\otimes \mathbb{N}}$ is defined as an inductive limit with connective maps $a \mapsto a \otimes 1$.

What are your main interests? I see you have added the "operator-algebras" tag, so maybe you should look into nuclearity and exactness of reduced group C*-algebras. The classical reference for this is Brown and Ozawa's book. If you, on the other side, are more interested in K-theory, the reasoning why property A and amenability play a role in the Baum-Connes conjecture is fundamental.

Awesome, thanks! I was wondering indeed that it should not be countable, glad to know for sure.

@Ycor Does it? I've been told otherwise. Do you have a proof of it?

If you don't care if $C$ depends on $n$, then yes. If you do care, then not.

I don't know about most of these questions, and don't know whether they are known or not, but $\mathbb{F}_\infty$ also has asymptotic dimension 1, as it's a tree.

As far as I understand it, the the K-theory of $A$ and $A \otimes \mathcal{Z}$ is the same. Therefore, if you want to classify C*-algebras via their K-theory, as the Elliott classification program does, you can only classify $A \otimes \mathcal{Z}$ instead of $A$. As $\mathcal{Z}$ is self-absorbing, you actually want $A$ to be $\mathcal{Z}$-stable.

It should be remarked that all the former mentioned theorems actually prove that the involved traces are quasi-diagonal, not only the C*-algebras.

The most general theorem so far (as far as I know) is the celebrated Tikuisis-White-Winter theorem (see "Quasidiagonality of nuclear C*-algebras", Ann. Math. 2017, jstor.org/stable/24906439). It says that nuclear UCT algebras with faithful traces are quasi-diagonal. You can even change nuclearity with exactness when the trace is amenable, see "Quasi-diagonal traces on exact C*-algebras", J. Func. Anal. 272, by Jamie Gabe. Or the alternative proof by Chris Schafhauser in "A new proof of the Tikuisis-White-Winter theorem", J. Reine Angew Math 2017.

Hm.. That's indeed what I suspected, but I hadn't thought of going that way. Thanks!