If $F$ is a sheaf on a topological space $X$, the well-known étale space
contruction gives rise to a bundle $\Gamma F$ on $X$ such that $F$ is
isomorphic to the sheaf of sections of $\Gamma F$.

On the other hand, if $\mathfrak X$ is a stack on $C$ the Grothendieck
contruction gives an equivalence between the category of sheaves on
$\mathfrak X$ and the category of faithful morphisms of stacks on $C$ with
target $\mathfrak X$.

My question is:

1: Is the Grothendieck construction a generalisation of the étale space
construction?

In that case we could try to obtain the analogue result in the context of
differentiables spaces or stacks:

2: If $\mathfrak X$ is now a differentiable space, is there an equivalence
between the cat of sheaves on $\mathfrak X$ and the cat of faithful
morphisms of differentiable spaces  with target $\mathfrak X$?

3: Same question for $\mathfrak X$ a differentiable stack.