I think, that the levelwise colimit agrees with the colimit in the category Func$(\Delta,\mathcal{C}$. As the colimit may be viewed as a functor from the category of directed systems over $\mathcal{C}$ to $\mathcal{C}$ the differentials/degeneracies in a directed system of objects in Func($\Delta,\mathcal{C})$ induces differentials/degeneracies on the levelwise colimit.

So the levelwise colimit is really a simplicial object. And then one can show, that it satisfies the universal property of the colimit in the category Func$(\Delta,\mathcal{C})$.

The same proof also works for limits, or any other construction (I can imagine) defined by an universal property, that doesn't involve the simplicial structure.

I think, there is also a more conceptual proof of this, but I can't remember.

However it is not true, that limits commute with geometric realization: Any set may be viewed as a constant functor $\Delta\rightarrow \mathcal{Sets}$. A levelwise inverse limit of constant functors is again a constant functor and its realization is discrete, while the inverse limit of the realizations need not be discrete.