Suppose that $K$ is an $E_\infty$-algebra on a space $X$ (more generally, any ringed topos; also, feel free to assume that $X$ is a point). That is, $K$ is a cochain complex of sheaves on $X$, endowed with a multiplication which is commutative and associative `up to coherent homotopy' (=an algebra structure over an $E_\infty$-operad). You can assume for simplicity that the sheaves $K^i$ and the cohomology sheaves $H^i(K)$ are flat (so we don't have to worry about derived vs usual tensor product). Over $\mathbb{Q}$, we know that every such $K$ is quasi-isomorphic to a commutative differential graded algebra (cdga), see e.g. 1, Part II, Corollary 1.5, but I'm interested in the integral case.

Question 1.Is there a known criterion for $K$ being quasi-isomorphic to a cdga?

For each prime $p$, $H^*(K)\otimes \mathbb{F}_p$ has a natural action of the mod $p$ Steenrod algebra (1, I 7). I think I might have read somewhere that if $K$ is quasi-isomorphic to a cdga, this action has to be trivial. Is that true?

Question 2.Is the vanishing of the Steenrod operations on $H^*(K)\otimes \mathbb{F}_p$ necessary and/or sufficient for the existence of a cdga quasi-isomorphic to $K$?

1 I. Kriz, J. P. May *Operads, Algebras, Modules, and Motives*, Asterisque 233, available online here.