Igor is giving a good reference to the topic. For completeness, my education, and satisfaction of other reader's laziness, I'm going to give a rough outline here.

1. A Poincaré complex is, very similarly to what I've conjectured, a finite CW complex together with a chosen fundamental class that induces Poincaré duality.

• Such a complex of course does not correspond to a unique topological (let alone smooth) manifold, and there is a popular 5-dimensional counterexample.
• There are also "Poincaré pairs", corresponding to manifolds with boundary, essentially answering half of my last question.

2. Another important construction that one can do with a manifold (indeed, any nice enough vector bundle) is the Thom-Pontryagin construction. It leads to something called the "Spivak normal fibration".

• Essentially, one takes a vector bundle $p\colon E \to M$, picks out the unit disk bundle and quotients by the unit sphere bundle (thus compactifying each fibre), yielding the Thom space $T(p)$.
• Now, one can also first embed $M$ (uniquely) into $\mathbb{R}^k$ for $k$ high enough, and compactify the complement of the stable normal bundle $\nu$. Since the 1-point-compactification of $\mathbb{R}^k$ is $S^k$, this gives a map $S^k \to T(\nu)$, the "normal invariant".
• One can generalise and axiomatise this construction for Poincaré duality spaces, which is then called a "Spivak normal fibration". It consists of a spherical fibration for the CW complex and something generalising the normal invariant.

3. It is a theorem by Spivak that every Poincaré duality space has a Spivak normal fibration, unique up to homotopy equivalence. This is remarkable, since a Poincaré duality space will in general not even have a tangent or normal bundle.

• When the spherical fibration actually comes from the stable normal bundle of a smooth manifold, it has a reduction to a $O(k)$ structure group. Alternatively, one can think about the classifying space $BG$ of circle bundles, and the classifying space $BO$ of vector bundles, which is mapped into $BG$ by means of the $J$-homomorphism. Only those spherical fibrations in its image can come from a smooth manifold.
• Similar arguments can be made for topological and $PL$-manifolds.

4. For high dimensions, all this is already pretty close to a full answer. Dimensions 3 and 4 are the hardest, as usual.

• In 2 dimensions, oriented closed surfaces are classified by Poincaré complexes. (It seems this was proved as late as in the early '80s.) That seems to answer my first question.
• There is a theorem in $\geq 5$ dimensions by Browder that asserts that a simply connected Poincaré complex is homotopy equivalent to a manifold iff the normal fibration has a $TOP$-reduction.
• In 3 dimensions, Poincaré complexes were classified by Hendriks and Turaev. But I don't know which ones correspond to manifolds, and what extra data classifies the manifolds. (One could ask e.g. how to distinguish homotopy equivalent Lens spaces. Of course I'm not asking you to directly classify 3-manifolds.)
• In 4 dimensions, simply connected topological manifolds are classified by the intersection form and the Kirby-Siebenmann invariant. The intersection form is obviously defined for any 4-dimensional Poincaré complex. I think Kirby-Siebenmann does as well, but I'm not so sure about that.

5. To really find a manifold from a Poincaré complex, one needs to study $L$-groups, which are a whole huge subject in itself. (See e.g. this introduction by Wolfgang Lück or this book by Andrew Ranicki, or Wall's "Surgery on compact manifolds".)

• The basic idea seems to be that one studies maps from a manifold into a Poincaré space, and then tries to change the manifold by surgery until the map is a homotopy equivalence. The $L$-groups give obstructions to this.