A commutative monoid with zero is a commutative monoid $A$ together with an element $0_{A}$ such that $0_{A}a=a0_{A}=0_{A}$ for all $a\in A$. They are precisely the monoids (in the sense of monoidal categories) in the category of pointed sets, equipped with the smash product.

Passing to the derived world, commutative monoids get replaced by $\mathbb{E}_{\infty}$-monoids in spaces, while pointed sets get replaced by pointed spaces. So the natural analogue of a commutative monoid with zero in homotopy theory should be an $\mathbb{E}_{\infty}$-monoid in the symmetric monoidal $\infty$-category $(\mathcal{S}_*,\wedge,S^0)$ of pointed spaces.

Questions:

1. Grouplike $\mathbb{E}_{\infty}$-spaces are equivalent to infinite loop spaces, which are in turn equivalent to connective spectra. Is there a similarly nice description of $\mathbb{E}_{\infty}$-monoids in pointed spaces, or of some subclass of them?
2. Has the notion of an $\mathbb{E}_{\infty}$-monoid in pointed spaces been studied before, particularly in the classical homotopy theory literature, and certainly under another name?
3. The $\infty$-category of $\mathbb{E}_\infty$-spaces admits a tensor product $\otimes_\mathbb{F}$, having unit the geometric realisation $$|\mathbb{F}|\cong\coprod_{n=0}^{\infty}\mathbf{B}\Sigma_{n}$$ of the groupoid of finite sets and permutations $\mathbb{F}$. Is there such a tensor product in the $\infty$-category of $\mathbb{E}_{\infty}$-monoids in $\mathcal{S}_*$, and, if so, what is its monoidal unit?
4. A natural example of an $\mathbb{E}_\infty$-monoid in pointed spaces is $\Omega^\infty E$ for $E$ any $\mathbb{E}_{\infty}$-ring spectrum, as pointed by Maxime Ranzi in the comments. What are some other nice examples?
5. The $\infty$-category of $\mathbb{E}_{\infty}$-spaces admits a number of point-set models, such as special $\Gamma$-spaces, commutative monoids in $*$-modules, and commutative monoids in $\mathcal{I}$-spaces. Similarly, the $\infty$-category of spectra has $\mathbb{S}$-modules, symmetric spectra, and orthogonal spectra as point-set models (there's also very special $\Gamma$-spaces for connective spectra). Does the symmetric monoidal (?) $\infty$-category $\mathsf{Mon}_{\mathbb{E}_{\infty}}(\mathcal{S}_*)$ also admit point-set models, in the sense of a presentation by a monoidal model category?