Suppose we are given a type $(0,2)$ tensor $T_{\mu\nu}$ in a Minkowski space with $(-,+,+,+)$ signature. Consider a closed 3-dimensional hypersurface $\partial \Omega$ which encloses a volume $\Omega$ in the Minkowski space-time. We have the following identity:$$\int_{\partial\Omega}T_{\mu\nu}n^\nu~dvol_3=0\tag{1}$$where $n^\nu$ is the contravariant normal vector pointing out of the hyperspace $\partial \Omega$. Then by the Divergence Theorem in higher dimensions, we could transform the above integral to $$\int_{\Omega}\text{div}~T_{\mu\nu}~dvol_4 \tag{2}$$
My questions are as follows:
Am I doing Gauss' Divergence Theorem (especially with regard to notation) correctly from equation (1) to equation (2)? I figure that as we have a 2-form in front of a contravariant, I might have made a mistake in keeping $T_{\mu\nu}$ as it is. Do we need to swap $T_{\mu\nu}$ to $T_\mu^\nu$ and why?
If I am doing (1) to (2) correctly, then what is the index notation expression for $\text{div}~T_{\mu\nu}$? I intend to say it is $\partial_{\nu}T_{\mu\nu}$ but it is in fact $\partial^{\nu}T_{\mu\nu}$. Why is this the case?