About the first question: Yes, the simplicial nerve is an instance of the same general construction which gives you the usual nerve (and e.g. also the Quillen equivalences between simplicial sets and topological spaces, between models for the $mathbb{A}^1$-homotopy category given by simplicial presheaves and sheaves respectively and much more):
A cosimplicial object in a cocomplete category E gives, via Kan extension, rise to an adjunction between $E$ and simplicial sets, where the left adjoint goes from simplicial sets to $E$ and comes from the universal property of the Yoneda embedding (namely that functors from $C$ to $E$, $E$ cocomplete, are the same as colimit perserving functors from $Set^{C^{op}}$ to $E$). This is (part of) Proposition 3.1.5 in Hovey's book on model categories.
Lurie uses the same pattern; it is enough to give a cosimplicial object in simplicial categories, which is done in his definition 1.1.5.1.
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