Local questions:

1) Given a commutative ring $A,$ is $Sh_\infty\left(Spec(A)\right)$ hypercomplete?

2) Given a commutative ring $A,$ is $Sh_\infty\left(Et\left(A\right)\right)$ hypercomplete, where $Et\left(A\right)$ is the small etale site?

3) and 4) The same questions but for $A$ a simplicial commutative ring

5) and 6) The same questions but for $A$ and $E_\infty$-ring spectrum

Or perhaps something general is known about when $Spec^{\mathcal{G}}\left(X\right)$ is hypercomplete, where $\mathcal{G}$ is a geometry in the sense of Lurie's DAG V?

"Large" Global questions:

Ignoring size issues (e.g. by using universes) are infinity sheaves on any of the following hypercomplete?

A) Affine schemes with the Zariski topology

B) Affine schemes with the Etale topology (or flat, etc.)

C) and D) same question for simplicial affine schemes

E) and F) same question for spectral schemes

(I have a feeling that the answer might depend on things being Noetherian)

Any references would be great also, thanks!