Is there an example of such a category that you are interested in?

It's not really clear what it means to "tell the bands apart". Continuous parametrisations of individual Eigenvalues of a family of matrices exist only locally (the corresponding bundle can be globally nontrivial), and as you observe, they are not well-defined around places with multiple Eigenvalues. So it isn't clear what even is meant by "the correct result".

In that case, the claim is that $H_*(C\otimes_k D) \cong H_*(C)\otimes_k H_*(D)$, for $C,D$ complexes of $k$-vector spaces, which is the classical Künneth theorem from homological algebra. You can prove it for example by observing that in $D(k)$, every object is equivalent to a sum of shifts of the trivial module $k$, so you can reduce to that case.

If I were you I would remove the bit about $M_n(\mathbb{Z})$, now it is really confusing to tell what you're looking for. If you're not after ring extensions and happy with some injective multiplicative monoid map from $\mathbb{Z}$ to some other ring $R$ which takes primes to irreducibles and hits all but finitely many irreducibles, you can take basically any $R$ with countably many irreducibles (for example $\mathbb{Z}$ itself).

I think the $M_n(\mathbb{Z}) $ example is flawed. The unique ring map from $\mathbb{Z} \to M_n(\mathbb{Z})$ takes $p$ to a determinant $p^n$ matrix ($p$ times the unit matrix), so the linked answer doesn't apply. And indeed, $5$ splits in $M_2(\mathbb{Z})$, since $M_2$ contains a copy of $\mathbb{Z}[i]$.

Torsion-free modules over PIDs are flat

Sure, it just gets annoying since your subring can't be a PID. I think something like the free associative algebra generated by $x, y, z$ subject to $xy=yx$ and $xz=y$ works (with subring $k[x, y]$) (but this is much larger than my original example and I haven't carefully checked it)

$(-)^\times$ is a pretty weird name for what seems to be the forgetful functor from abelian groups to sets...

The correct notion of object with (coherent) involution (i.e. $C_2$ action) in an $\infty$-category is still a functor $BC_2\to \mathcal{C}$.

I think statistically the statement is correct - there exist many good programmers who don't know schemes ;)

The Intuition from the plane doesn't carry over: the plane can be tesselated with equilateral triangles, and a hexagon can be decomposed in 6 equilateral triangles. There is no tesselation of 3d space into equilateral tetrahedra. Somewhat related, the icosahedron doesn't decompose into 20 equilateral tetrahedra. (For example, the radius of the circumscribed sphere of the icosahedron does not agree with the edge length, contrary to what happens for the hexagon)

Superficially, it looks like you want a natural transformation between functors $\operatorname{Man} \to \operatorname{Cat}$. However, for that to make sense, you need to make sense of $\Omega(M)$ as a category, and pointwise $T(M): \operatorname{Vect}_\Delta(M) \to \Omega(M)$ as a functor of categories.

As an explicit example, I like $SO(3)$ with $a,b$ two rotations by the same angle around two different axes which are very close to each other. For any $m$, $a^m$ and $b^m$ will still be two rotations by the same angle around those axes, thus close to each other.