One of the standard proofs of Poincaré duality, at least for those manifolds that have handle decompositions, provides a reason for some of these naturality properties. Every piecewise linear manifold, or every smooth manifold, has a handle decomposition, and many but not all topological manifolds also do. (Amazingly enough, the only exceptions are in 4 dimensions.) A handle decomposition gives rise to two different CW cellulations on the manifold, one using cores and the other using co-cores. Then this proof of Poincaré duality posits that the CW chain complex of one cellulation is identical to the CW cochain complex of the other cellulation.
You can extend this coincidence of chain complexes to both of your examples, the Mayer-Vietoris sequence and the exact sequence of a pair. Obtaining identical chain complexes also gives you other information, for instance that the Bockstein maps are the same.
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:
http://arxiv.org/abs/math/0311016.
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:
http://arxiv.org/pdf/math/0309455v2
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:
http://arxiv.org/abs/0710.3550
That ended up being a long answer! Please ask if something isn't clear!
Best Answer
1. Introduction
2. Easy example
3. Normality, Duality
4. Normal example
Introduction
The pseudomanifold and homology manifold conditions are both local conditions, while Poincaré duality is a global condition. It is possible for a pseudomanifold to fail the homology manifold conditions in several places, but so that the local deviations globally cancel, so it retains duality.
Simplicial complexes are "locally cone-like": every point $x$ has a neighborhood that is the cone on another space $L$, the link, with $x$ the cone point. Then $$H_*(X,X-\{x\})=H_*(CL,L\times I)=\tilde H_*(SL)=\tilde H_{*-1}(L)$$ So the homology manifold condition is that the links have the homology of spheres.
Easy example
The interval admits an involution reversing the endpoints. The quotient by this involution is again contractible. Think of this quotient operation as gluing one half of the interval to the other half. Embed the interval in an $n$-manifold. Form a quotient of the manifold by gluing the one half of the interval to the other half. If $n\geqslant 3$, this leaves the cells in degree $n$ and $n-1$ unchanged, so does not affect the pseudomanifold condition. Since we have replaced one contractible subspace by another, the homotopy type of the total space is unchanged and satisfies duality (with the pseudomanifold fundamental class, etc). But the the links are no longer spheres at any point along the interval. At the image of the ends of the interval, the link is two spheres glued in one point. At general points along the interval, it is two spheres glued in two points. At the image of the midpoint, the fixed point of the involution, the link is a single sphere with two of its points glued to each other.
Normality, Duality
That example involved changing a manifold in low dimensions, which doesn't violate the high dimensional pseudomanifold condition. To rule out such examples, one has the concept of a normal pseudomanifold (cf normal variety). The normalization of a pseudomanifold is produced by taking a disjoint union of $n$-simplices parameterized by those of the original space and gluing them along their $n-1$-faces, according to how the original was glued. Thus the normalization of the above example is the unmodified manifold. A normal pseudomanifold is one isomorphic to its normalization.
It may be useful to use the Verdier dualizing sheaf. The cohomology with coefficients in the dualizing sheaf matches the homology: $H^*(X; D)\cong H_*(X)$. An oriented pseudomanifold yields a map of sheaves $\mathbb Z\to D$. We seek Poincaré duality, that is, $H^*(X; \mathbb Z)\cong H_*(X)$. So we seek spaces where the map of sheaves induces an isomorphism on cohomology $H^*(X; \mathbb Z)\cong H^*(X; D)$, even though it is not an isomorphism of sheaves. That is, we seek the cone to be a nontrivial sheaf with no cohomology in any dimension. There are such sheaves, such as a local system supported on a circle with appropriate monodromy. That motivates the construction. It also gives immediate proofs of the claims, but it should not be hard to check them without using sheaves.
Normal example
Take an $n$-manifold manifold $M$ and an automorphism $\phi$ so that the action of on the homology in intermediate degrees is sufficiently mixing so that if we consider it an action of the group $\mathbb Z$ and take group homology with those coefficients, the homology is trivial. For example, $M=T^2$ and $\phi=\left(\begin{matrix}2&1\\1&1\\ \end{matrix}\right)$. Then form the mapping torus ($T_\phi=M\times I/(x,0)\sim(\phi(x),1)$). This is an $M$-bundle over the circle, homology computed by a spectral sequence $H_*(S^1;H_*(M))\Rightarrow H_*(T_\phi)$. By assumption on $\phi$, the spectral sequence degenerates and $H_*(T_\phi)=H_*(S^1\times S^n)$. Then cone off each copy of $M$, forming a circle of cone points. To put it another way: form the mapping cylinder of the map to the circle $T_\phi\to S^1$. Yet another way: the mapping torus of the self-map of the cone $\tilde\phi\colon CM\to CM$. This space is a normal pseudomanifold (with boundary) because those properties are preserved by cones and products. Its set of singular points is a circle, and the dualizing sheaf twists about it with the prescribed monodromy, so it contributes nothing to the sheaf cohomology. If you prefer a closed example, double the space along the boundary (or equivalently take the double mapping cylinder of the map to the circle; or the mapping torus of the automorphism of the suspension). This gives a pseudomanifold that is not a homology manifold, but which satisfies Poincaré duality. Just as the mapping torus had the homology of a manifold $S^1\times S^n$, this space has the homology of $S^1\times S^{n+1}$. Actually, that is only correct if we restrict to trivial coefficients. If we define Poincaré duality to be for all local systems, then this space fails, for some unwind the twist around the singular circles. However, we can eliminate them by killing the fundamental group by surgery. That is, cut out a neighborhood of a circle, $S^1\times D^{n+1}$, leaving a boundary $S^1\times S^n$ and fill it in with $D^2\times S^n$. Now local systems are trivial, so it satisfies full Poincaré duality.