Volume Integral : $\int \mathrm d^3\mathbf{r}' \frac{\nabla \cdot \mathbf{M}(\mathbf{r'})}{|\mathbf{r'}-\mathbf{r'}|}$

Q1: In my Classical Electrodynamics book (Jackson), the derivation is a little different. By localized in Jackson, he means that there exists an object with magnetization within some fixed boundary, and $M(r) = 0$ outside (and on) that boundary. So the first integral is $\int_V \nabla \cdot \frac{M(r')}{|r-r'|}$ dV, right? The divergence theorem then says this integral is equal to $\int_s \frac{M(r')}{|r-r'|} \cdot n\, dS$ . The idea then is that M(r') vanishes on this boundary. If we want to go a little deeper, I believe you can choose the volume of integration $V$ s.t. the surface of $V$ lies outside the magnet where $M(r) = 0$.

See Classical Electrodynamics, John David Jackson. Wiley. This has the answer to Q2 as well.