Yes, so:

A simplicial set is indeed precisely a presheaf on the simplex category.

There are various model category structures on categories of presheaves in general and on simplicial sets in particular.

With respect to the standard model structure on simplicial sets the Kan complexes are precisely the fibrant objects.

With respect to the local model structure on presheaves on a site the sheaves are precisely the fibrant objects.

There is a very useful combination of these two statements:

A simplicial presheaf is a presheaf on the product category of the simplex category and some site.

in the local projective model structure on simplicial presheaves the fibrant objects are precisely those simplicial presheaves that are Kan-complexes over each object of the site and that satisfy the oo-version of the sheaf condition ("descent"): these are the oo-stacks on the given site.

Yes, so:

A simplicial set is indeed precisely a presheaf on the simplex category.

There are various model category structures on categories of presheaves in general and on simplicial sets in particular.

With respect to the standard model structure on simplicial sets the Kan complexes are precisely the fibrant-and-cofibrant objects.

With respect to the local model structure on presheaves on a site the sheaves are precisely the fibrant-and-cofibrant objects.

There is a very useful combination of these two statements:

A simplicial presheaf is a presheaf on the product category of the simplex category and some site.

in the local projective model structure on simplicial presheaves the fibrant objects are precisely those simplicial presheaves that are Kan-complexes over each object of the site and that satisfy the oo-version of the sheaf condition ("descent"): these are the (hypercomplete) oo-stacks on the given site.