Are these conditions sufficient to calculate this expectation? We have
\begin{aligned}
E(Z_1) = A \\
\Pr \{ Z_2 = Z_1 + 1 \} = \frac 1 2  \\
\Pr \{ Z_2 = Z_1 - 1 \} = \frac 1 2 
\end{aligned}
Are these conditions enough to get $E(Z_2)$? 

Yes, because $\newcommand{\E}{\mathbf{E}}$
$$
\E[Z_2] = \E[Z_1] + \E[Z_2-Z_1]
$$
holds for all random variables $Z_1$ and $Z_2$ (and you can calculate each term on the right). In other words, linearity of expectations can be applied in all cases, however correlated the random variables may be. 
