Structure-preserving maps: homomorphisms and isomorphisms

Which of these maps $f : \Z \to \Z$ preserve addition, i.e. satisfy $f(x + y) = f(x) + f(y)$ for all $x, y$? (Choose all that apply.)

Use reduction modulo $9$ without multiplying out: $47 \equiv 2$ and $85 \equiv 4 \pmod 9$ (digit sums $11 \to 2$ and $13 \to 4$). What is the remainder of $47 \times 85$ on division by $9$?

$\exp : (\R, +) \to (\R_{>0}, \times)$ is an isomorphism. Under it, the additive inverse $-x$ of $x$ corresponds to which element of $\R_{>0}$?

How many homomorphisms $(\Z/4, +) \to (\Z/6, +)$ are there?

Why is $(\Z/4, +)$ not isomorphic to the symmetry group of a non-square rectangle?

In the proof of Theorem 3, $\; g(u \star v) \overset{(1)}{=} g\big(f(g(u)) \star f(g(v))\big) \overset{(2)}{=} g\big(f(g(u) \ast g(v))\big) \overset{(3)}{=} g(u) \ast g(v)$. Which step uses that $f$ is a homomorphism?

Which of these is a structure-preserving bijection whose inverse is not structure-preserving?

Which Lean proposition says that f : ℤ → ℤ preserves addition?

Write preserves_addition(f, m, n), where f is a list of length m with f[k] in range(n), describing a map $\Z/m \to \Z/n$. Return True if $f((x + y) \bmod m) = (f(x) + f(y)) \bmod n$ for all $x, y$, else False. Then use it to count the homomorphisms $\Z/6 \to \Z/4$ (the tests do this for you).

The map $\Z/6 \to \Z/2 \times \Z/3$, $x \mapsto (x \bmod 2, \; x \bmod 3)$ is an isomorphism (of rings, even). Which $x \in \Z/6$ is sent to $(1, 2)$?