Skip to the bottom for my questions, first some discussion:
It is a celebrated theorem of Kadeišvili that $A_{\infty}$-algebra structures can be transferred along homotopy equivalences so that the homotopy equivalences are $A_{\infty}$ maps. There are some results in this direction where 'homotopy equivalence' is replaced by 'quasi-isomorphism', for example Burke shows in Transfer of A-infinity structures to projective resolutions that under some mild hypotheses if $R$ is a $Q$-algebra then a projective $Q$-resolution $P_{\bullet}$ of $R$ has an $A_{\infty}$-algebra structure so that the map $P_{\bullet} \to R$ is $A_{\infty}$.
I am interested in these types of results for $E_{\infty}$-algebras in chain complexes, and after looking at the literature I am a little confused. First, I believe that there can't be such a general result for $E_{\infty}$-algebras because if $X$ is any space then the singular cochain complex $S^{*}(X,k)$ over a field $k$ is an $E_{\infty}$-algebra that is homotopy equivalent to its cohomology algebra $H^*(X,k)$ (which as a graded commutative ring is also an $E_{\infty}$-algebra). The homotopy equivalence $S^{*}(X,k) \to H^*(X,k)$ can't be an $E_{\infty}$ map because of the action of the Steenrod squares-- the homology of any $E_{\infty}$-algebra has Steenrod operations, but all operations except for $Sq^j$ on classes of degree $j$ are zero for a strictly commutative algebra.
However, I have seen more general theorems that appear to give as a special case that $E_{\infty}$-algebra structures can always be transferred along quasi-isomorphisms. For example, in Axiomatic Homotopy Theory for Operads, Theorem 3.5, I understand Berger and Moerdijk to state that :
If $f:X \to Y$ is a weak equivalence between bifibrant objects in a monoidal model category $\mathcal{M}$ where the operads carry a transferred model structure and $\mathcal{P}$ is a cofibrant operad, then any $\mathcal{P}$- algebra structure on $X$ (resp. $Y$) induces a $\mathcal{P}$-algebra structure on $Y$ (resp. $X$) in such a way that $f$ preserves the $\mathcal{P}$-algebra structures up to homotopy.
(In the topological setting this theorem is the 'Homotopy Invariance Property' of Boardmann and Vogt)
This theorem seems to be in contradiction with the map $S^{\bullet}(X,k) \to H^*(X,k)$ not being $E_{\infty}$--I believe that chain complexes of $k$-modules with the projective model structure for $\mathcal{M}$ and an $E_{\infty}$ operad for $\mathcal{P}$ satisfy the necessary hypotheses.
So, my questions are:
Question One: Working with chain complexes of $Q$-modules (where $Q$ is a commutative ring), given a quasi-isomorphism $X \to Y$ under what conditions does an $E_{\infty}$-algebra structure on $Y$ lift to a compatible one on $X$?
Question Two: If you cannot always compatibly lift an $E_{\infty}$-algebra structure, what is lacking with chain complexes and $E_{\infty}$ operads in order to apply the homotopy invariance property?
Question Three: If you can always lift an $E_{\infty}$-algebra structure, how does this work for the homotopy equivalence between the singular chain complex of a space and its cohomology?
