Which simplicial objects are Čech nerves?

The analogous 1-categorical version of your first question would be "which parallel pairs are kernel pairs?" As far as I know this does not have a non-tautological answer in an arbitrary category, but in a Barr-exact (= effective regular) category the answer is "the internal equivalence relations" (although this is more or less part of the definition of "Barr-exact", so it may not be very satisfying). The $\infty$-version of an internal equivalence relation is an internal groupoid object, and the corresponding equivalence can be found for $\infty$-toposes and similar categories in Higher Topos Theory, though I don't know whether an exact $(\infty,1)$-categorical analogue of "Barr-exact" has been defined yet.

The answer to your second question is also yes in good $(\infty,1)$-categories such as $\infty$-toposes (and presumably also "Barr-exact" ones, whatever those are), because in that case the effective epis are the left class in a factorization system whose right class are the monomorphisms, and it's easy to see that any $(\infty,1)$-regular epi is left orthogonal to monomorphisms, hence is effective. Here, of course, the analogous 1-categorical question has the answer "yes" in all categories, not just the Barr-exact ones, and I don't know whether that is still true in the $\infty$-case.