Why does exponentiating group elements by integers always make sense? Let $G$ be a group.
For any $g\in G$, we can define a mapping $\mathbb Z\to G$ via $n\mapsto g^n$.
This map is a homomorphism between the additive group $\mathbb Z$ and $G$.
On the one hand, it's obvious why we can always do this. We just define $g^n$ as "multiply $g$ with itself $n$ times" (for $n>0$, and then straightforwardly generalise to all $n\in\mathbb Z$).
Thus, $\mathbb Z$ comes into play because we are "counting" the number of times $g$ is operating on itself.
On the other hand, this means that for any group $g\in G$, there is a homomorphism $\mathbb Z\to G$ sending $n$ to $g^n$.
Is there any reason why this group in particular, $\mathbb Z\equiv(\mathbb Z,+)$, ends up being homomorphic to subgroups of every group? Are there groups other than $\mathbb Z$ sharing this property?

First, for each $g$ your map is really a homomorphism of the integers $\mathbb{Z}$ to $G$. You map $-1$ to $g^{-1}$. You don't want to restrict the map to the positive integers.
All you are really saying is that any group has lots of cyclic subgroups - every element generates one. Any any cyclic group is a homomorphic image of $\mathbb{Z}$.
You can always map any group $H$ homomorphically to $G$ with the trivial map the sends everything to the identity.
