It is well known (and asked countless times as I gathered while searching an answer for this question) that $\mathbb{Z}/n\mathbb{Z}$ $\cong$ $\mathbb{Z}_n$. However, what I am trying to do is to infer the structure of $\mathbb{Z}/\mathbb{Z}_n$ (if this is even a valid structure that is, as I am not even sure that $\mathbb{Z}_n$ is a valid normal subgroup of $\mathbb{Z}$ under normal addition: details below).
Obviously taking a quotient group is not the same thing as division as we are used to (as in multiplying by the multiplicative inverse), but the separation (partition) into "equal" groups of the original supergroup does resemble similar concepts. I was wondering if this structure will also be isomorphic to $n\mathbb{Z}$ as if we simply switched the denominator like in regular division. I simply cannot comprehend how to partition $\mathbb{Z}$ by the cosets of $\mathbb{Z}_n$.
And this ultimately is my problem. If I take regular (nonmodular addition) clearly if I take the coset where I add 1 to, say, $\mathbb{Z}_5$, this overlaps with $\mathbb{Z}_5$, and does not parition $\mathbb{Z}$. But then I also consider how nonmodular addition will not make $\mathbb{Z}_5$ a well defined binary structure in the first place. So if we consider just modular addition we arrive that it is its only coset.
So did I answer my own question? The only reason that we can't have cosets of $\mathbb{Z}_n$ in the context of a supergroup $\mathbb{Z}$ is because of the fact that their operations are incompatible, and would not even make sense, as the incompatibility would not make it a proper subgroup. I am curious as to the relationship here as it has never been brought up in any introductory algebra class.
I think you are having notation issues. The notation $\mathbb{Z}/n\mathbb{Z}$ always means the integers modulo $n$. The notation $\mathbb{Z}_n$ is sometimes used sometimes for the integers modulo $n$, but sometimes it is not. See the section of the wikipedia on notation.
However the way you are using it means the integers modulo $n$. Same object, different notation.