Yes, $\Sigma^\infty_+:\mathrm{Space}→\mathrm{Spectra}$ is a symmetric monoidal left adjoint, so it can be extended to an adjunction $$\Sigma^\infty_+:\mathrm{CAlg}(\mathrm{Space})→\mathrm{CAlg}(\mathrm{Spectra})\dashv \Omega^\infty:\mathrm{CAlg}(\mathrm{Spectra})→\mathrm{CAlg}(\mathrm{Space})$$ (see Higher Algebra, 7.3.2.13). The functor $\Sigma^\infty_+$ is what you want to call $\mathbb{S}[-]$.

The answer to your question 2 is also true, since $\mathrm{Fun}(BM,\mathrm{Spectra})$ has a compact generator ($\mathbb{S}[M]$ with the obvious $M$-action, which represents simply the evaluation at the basepoint of $BM$) and then you can just apply Schwede-Shipley Morita theory.

I think you made a sign mistake, and asked for a left adjoint to $\Omega^\infty$ since the monoid ring is a left along to the forgetful functor.

If so then the answer is yes. $\Sigma^\infty_+:\mathrm{Space}→\mathrm{Spectra}$ is a symmetric monoidal left adjoint, so it can be extended to an adjunction $$\Sigma^\infty_+:\mathrm{CAlg}(\mathrm{Space})→\mathrm{CAlg}(\mathrm{Spectra})\dashv \Omega^\infty:\mathrm{CAlg}(\mathrm{Spectra})→\mathrm{CAlg}(\mathrm{Space})$$ (see Higher Algebra, 7.3.2.13). The functor $\Sigma^\infty_+$ is what you want to call $\mathbb{S}[-]$.

The answer to your question 2 is also true, since $\mathrm{Fun}(BM,\mathrm{Spectra})$ has a compact generator ($\mathbb{S}[M]$ with the obvious $M$-action, which represents simply the evaluation at the basepoint of $BM$) and then you can just apply Schwede-Shipley Morita theory.