Reflexive coequalizers are examples of sifted colimits in a 1-category, and groupoids are examples of reflexive coequalizers. But quotients don't preserve monomorphisms in general. 

This gives a counterexample to question (2) in the category Set: 

Let $V = S \rightrightarrows S$ denote the trivial groupoid acting on a set $S$ with two elements, and $W = S\times \mathbb Z/2\mathbb Z \rightrightarrows S$ the action groupoid of $\mathbb Z/2\mathbb Z$ acting on the same set $S$. There is an inclusion map $V\to W$, which is injective on both objects and morphisms. But the induced map on the quotient sets is not injective.

----

In the $(2,1)$-category setting, according to this [answer][1] of Denis Nardin, one must truncate the simplicial diagram at the 2-simplices to get a sifted diagram (note that a reflexive coequalizer diagram is the simplicial diagram truncated at the 1-simplices).

To get a counterexample to question (1), first take the corresponding 2-truncated Cech simplicial sets of the diagrams above, i.e.

$$S \rightrightarrows^{\rightarrow} S \rightrightarrows S$$

and

$$S\times \mathbb Z/2 \times \mathbb Z/2 \rightrightarrows^\rightarrow S\times \mathbb Z/2 \rightarrow S$$

Then take the category of sheaves (of vector spaces, say) together with the pushforward functor on these diagrams of sets (thought of as discrete spaces), to get truncated simplicial diagrams of categories (all of which are direct sums of copies of Vect). 

The diagrams are levelwise fully faithful, but the induced functor on the colimit is identified with the pushforward $Shv(S) \to Shv(pt)$ which is not fully faithful.


  [1]: https://mathoverflow.net/a/236600/43054