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The following is not exactly an answer to your questions, but maybe relevant nonetheless:
On 1): Without looking into details of the formulas given, one can say that the iteration of the plus constructions must agree, up to equivalence, or else there would be something wrong in their definitions. Namely, the iterated plus construction is meant to model the $\infty$-stackification (the 2-stackification in the case of NS’s article) in either case.
On 2): Just to remark that passing from the site of smooth manifolds to that of action groupoids by finite groups gives the “singular-smooth” homotopy theory which we discuss in Proper Orbifold Cohomology and in Equivariant principal $\infty$-bundles. There we discuss $\infty$-stackification not via the plus-construction but via special-purpose tools that apply thanks to the special nature of the site of smooth manifolds. (Essentially: Once a simplicial presheaf satisfies descent over just Cartesian spaces ,which is a rather more simple condition, then its stackification over all manifolds may be computed by evaluating it on Čech nerves of good open covers, which is typically exactly what one wants to compute anyways. The analogous statement holds also for the equivariant version over a site of finite action groupoids.)
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