Is this proper notation: $\;\mathbb Z^{+-}\;$? I have a question about notation. Is it okay to represent $\;\mathbb Z^{+-}$ as $ \mathbb Z - \{0\}$ ? 

Like does it make sense to manipulate both sets mathematically like that? ) i.e. $ \{0\} = \mathbb Z - \mathbb Z^{-+} $

Thanks in advance!
Using $\mathbb Z^{+-}$ would have given me little clue as to what you are denoting, were it not for the equality you use to define it. (Okay, I could have probably guessed, but why make your readers have to guess?)

I would strictly use $ \mathbb Z - \{0\}$, or better yet, $\mathbb Z \setminus \{0\}$. I'd even prefer it were spelled out in words: "all non-zero integers" than trying to guess what your notation denotes. (I have never seen the notation $\mathbb Z^{+-},\,$ to be honest.)

I don't think it makes sense to write $ \{0\}$ as $\mathbb Z - \mathbb Z^{-+} $. (I'm not clear why you'd want to avoid using $\{0\})$. In any case, using $\;\{0\}$ is much more straightforward, and easier to write and read.
$\Bbb{Z}-\{0\}$ is usually denoted by $\Bbb{Z}^*$, the set of invertible integers.