Are there general principles when a simplicial object is a (co)limit of other simplicial objects level by level, but is not a (co)limit when considering the entire simplicial structure?
Objects can be groups or algebras etc, and limits here can something as simple as products, coproducts or limits, colimits.
Limits and colimits can be computed point-wise in functor categories, such as the category of simplicial objects of any category. That is, if you have a functor category $\mathcal{C}^I$ ($I = \Delta$ in your case) and a functor $F: P \longrightarrow \mathcal{C}^I$, in particular, for every object $p$ in $P$ you have a functor $F_p : I \longrightarrow \mathcal{C}$. Then $\varinjlim_p F_p$ is an object of $\mathcal{C}^I$; that is, a functor $I \longrightarrow \mathcal{C}$, whose value on objects $i$ in $I$ is
$$ (\varinjlim_p F_p)(i) = \varinjlim_p (F_p(i)) \ . $$
This is true as far as the colimit on the right exists for every object $i$ in $I$. An analogous statement applies for limits in functor categories.
For limits, you can find the result in MacLane, "Categories for the working mathematician", chapter V, section 3, theorem 1 (see at the end of the proof also). For colimits, see "Sheaves in geometry and logic: a first introduction to topos theory", by Saunders Mac Lane,Ieke Moerdijk, p.40.
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.
