Let $\mathcal{M} \equiv (\mathbb{N}, 0, S, <, +)$ and consider the equivalence relation $\sim$ defined in $M$ by: $a \sim b$ if and only if $d(a, b) < \infty$, i.e. the distance is finite.
It is easy to check that the class $[0^\mathcal{M}]_\sim$ with the restricted operations and the restricted order of $\mathcal{M}$ is isomorphic to $(\mathbb{N}, 0, S, <, +)$, while for any $a \in \mathcal{M}$ not equivalent to $0^\mathcal{M}$ the corresponding equivalence class $[a]_\sim$ with the restricted operations and order is isomorphic to $(\mathbb{Z}, S, <, +)$, it is a $\mathbb{Z}$-copy.
Prove that the order $<_\sim$ of the $\mathbb{Z}$-copies of $\mathcal{M}$ is a dense linear ordering without endpoints.
I think that, given some element $a\in M$ in a $\mathbb{Z}$-copy, I need to consider in which copy should $na$ be for $n \geq 2$. But I don't visualize this problem well. Any help on how to start this would be appreciated.
Hint: as suggested by Asaf, the function $f(x) =\lfloor x/2\rfloor$ is definable. For density (and having no initial points), consider $f(x+y) $ (it's approximately the midpoint between $x$ and $y$). For having no largest element, use the function $x\mapsto 2x$, as you have said.
Using the isomorphism you have described, you can assume without loss of generality that $\mathbf N$ is contained in $M$. This should make things easier to visualise.