Jeff, your questions were in some sense the motivation for my thesis. Let me say a few things that you probably already know before I try and answer your questions.
The $E_\infty$ algebra structure on integral cochains of a topological space $X$ is a homotopy invariant of $X$. If $X$ is nilpotent and of finite type, then the quasi-isomorphism type of the cochain algebra is a complete homotopy invariant. This is a theorem of Mandell:
It is also true that the $C_\infty$ multiplication on cochains is a complete invariant of the rational homotopy type of a simply connected space, but I don't know of a place where this is written down explicitly. The problem is that Quillen and Sullivan wrote their papers before ideas like infinity algebras and Kozul duality were part of the general consciousness of topologists. However, Quillen shows in Rational Homotopy Theory that the (cocomutative) coalgebra on chains is a complete invariant of the rational homotopy type of a simply connected space. He goes to some trouble to construct a cocomutative coalgebra; nowadays we would say that he is just constructing a particular representative of the quasi-isomorphism type of cochain $C_\infty$ coalgebra which happens to be strictly associative.
Now, let me try and restate Jeff's question 1, which makes sense over Q or over Z. Fix a simply connected integral or rational PD space $X$. We know:
1) The cochain algebra $C^*(X)$, considered as an E or C infinity algebra on the integral or rational cochains, is a complete integral or rational homotopy invariant given restrictions on the fundamental group.
2) The homotopy class of the map $\mu_X: X \to BG$ which determines in the Spivak normal fibration is a homotopy invariant of $X$. (I don't actually know much about the rational version of this statement, but it looks like it's laid out in Su's thesis linked above.)
3) The topological (or smooth or PL) structure set of manifold structures in the homotopy type of $X$ is again a homotopy invariant of $X$.
I interpret Jeff's question 1 to be the following: the cochain algebra knows the all the homotopy invariant information about $X$, so how do we see the info of (2) and (3) as features of the cochain algebra? The problem is that the cochain algebra depends only on the homotopy type of $X$ as a $space$, not as a Poincare duality space. I don't think that that you can ever see, for example, the structure set from only the higher homotopies of the cup product. (Though I don't have a formal proof that it's impossible.)
If you want to detect manifold structures, you instead need to look at the Poincare duality map. In my thesis, I explain how you can write down Ranicki's total surgery obstruction -- which detects whether or not the structure set is empty -- as an obstruction to the existence of "local" inverse to the Poincare duality map. (This statement is over Z. I don't know of a rational version of Ranicki's total surgery obstruction, and Ranicki told me he doesn't either.)
Thus, as I understand it, the higher multiplications are not exactly the right place to look for obstructions to manifold structures; you need to look instead at the inverse of the Poincare duality map.
I know of a couple of different answers to Jeff's question 2 about the relation between the colagebra and the algebra structure on cochains.
1) There is the following paper of Tradler and Zeinalian:
One result of this paper is that the rational chains of a PD space form an $A_\infty$ coalgebra with an "$\infty$ duality". Presumably there is a dual statement for the cochains.
2) David Chataur has a result that for any PD space X, the PD map determines an equivalence of the cochains and chains of X as $E_\infty-C^*(X)$ modules. He sent me a sketch of the proof of this statement but I don't have his permission to disseminate it.
Morally, the answer should be that the PD map is an equivalence of "infinity Frobenius" algebras. Unfortunately there are many different definitions of Frobenius algebra, and there are technical problems with writing down the infinity versions of some these algebraic structures. (The ones that have a unit and a counit.) However see this paper of Scott Wilson's:
That ended up being a long answer! Please ask if something isn't clear!