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

For example, in the category Set (considering question (2)), 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.

I am not sure what this says about the categorical version, but I think it indicates that the sifted case is very different to the filtered case.

Edit: I think you can get a counterexample to (1) by taking categories of sheaves on the discrete sets above. by first taking 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 taking the category of sheaves (of vector spaces, say) on these discrete sets, equipped with the pushforward map, 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 colimit is identified with the pushforward $Shv(S) \to Shv(pt)$ which is not fully faithful.

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 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.

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

For example, in the category Set (considering question (2)), 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.

I am not sure what this says about the categorical version, but I think it indicates that the sifted case is very different to the filtered case.

Edit: I think you can get a counterexample to (1) by taking categories of sheaves on the discrete sets above.

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

For example, in the category Set (considering question (2)), 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.

I am not sure what this says about the categorical version, but I think it indicates that the sifted case is very different to the filtered case.

Edit: I think you can get a counterexample to (1) by taking categories of sheaves on the discrete sets above. by first taking 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 taking the category of sheaves (of vector spaces, say) on these discrete sets, equipped with the pushforward map, 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 colimit is identified with the pushforward $Shv(S) \to Shv(pt)$ which is not fully faithful.

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

For example, in the category Set (considering question (2)), 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.

I am not sure what this says about the categorical version, but I think it indicates that the sifted case is very different to the filtered case.

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

For example, in the category Set (considering question (2)), 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.

I am not sure what this says about the categorical version, but I think it indicates that the sifted case is very different to the filtered case.

Edit: I think you can get a counterexample to (1) by taking categories of sheaves on the discrete sets above.