What's "$\mathbb Z = (1)$ is cyclic"? As I understand it, $(1)$ is a cyclic group with $1$ being its generator. So, $1^n$ with $n \in \mathbb N$ generates $\mathbb Z^+$ , but what about $\mathbb Z^{-1}$ ? Do we say $(1)$ is a cyclic group with two operations $(+, -)$ defined on it so that $\mathbb Z = (1)$? 

In algebra, when you say that a set $S$ generates some structure $A$, you mean that any element of $A$ can be represented as repeated applications of operations of the structure in question to elements of $S$.
In your case, the structure is a group. A group has not only the binary operation, but also the inverse (and, technically, the identity element, considered as a nullary operation). So, $\{1\}$ indeed generates $\mathbb{Z}$, since the inverse of it (which is $-1$) and powers of inverse are also allowed.
