The solution of spin-1 massive field equation I am looking at chapter XI of Luke's QFT notes, where he has stated that the field satisfies the KG equation
\begin{equation}
(\Box+m^2)A_\nu=0
\end{equation}
And then he proceeds to define 
\begin{equation}
A_\nu=\sum_{r=1}^3\int\frac{d^3k}{(2\pi)^{3/2}\sqrt{2w_k}}\left(a_k^r\epsilon_k^re^{-ikx}+a_k^{r ^\dagger}\epsilon^{r*}_k e^{ikx}\right)
\end{equation}
For whatever reason, I am havng trouble seeing how this satisfies the above equation. I  thought that we were also under the condition that $\partial_\nu A^\nu=0\rightarrow \Box A_\nu=0$ Which means that m=0 as well, but it is massive. 
 A: To answer your first question: the expression that you have written pretty much satisfies the equations of motion, provided that
$$k ^{\mu} = \left\{ \sqrt{\vec{k}^{\,2} + m^2};\; \vec{k} \right\}. $$
I advice you to carry out the proof.
To answer your second question: no, we only use that condition to gauge-fix a theory with a gauge redundancy, and the theory under consideration is not gauge-invariant, because the mass term is not gauge-invariant.
UPDATE: actually I am mistaken, the Lorentz condition $\partial_{\mu} A^{\mu} = 0$ is used when deriving the equation of motion for the massive field. But nevertheless, the gauge invariance is gone. Thanks to J. Pak for the correction.
A: About your second question, I should correct the answer of 
Solenodon Paradoxus. Your spin-1 massive field equation is in fact the Proca equation which has been obtained by gauge condition: $\partial_\nu A^\nu=0$ . This equation is just closely related to the KG equation. Please see the Proca equation for detail.
