I've been interested in the possible (singular) cohomology groups of complex projective algebraic varieties, and there are lots of theorems that give various restrictions on these (Hodge decomposition, Lefshetz theorem, ... ). I realized that I would like to know more about the what is true for smooth manifolds hence my questions:

1.) One can construct CW complexes that have prescribed (reduced) homology groups (coeffs in $\mathbb{Z}$), these are the Moore spaces. However, they aren't even topological manifolds in general. Can one construct compact oriented smooth manifolds that have prescribed singular cohomology groups $H^i(X, \mathbb{Z})$, provided that after we remove torsion our sequence of groups satisfy Poincare duality? Should one expect that this is "generally possible" but it may be hard to actually construct examples?

2.) If $X$ is a compact oriented smooth manifold, is there any regularity in the torsion subgroups of it's cohomology: $H^i_{sing} (X, \mathbb{Z})$? (Eg, poincare duality gives regularity between the various torsion free parts.) How about if $X$ is a nonsingular complex projective variety?

3.) For $X$ a smooth oriented manifold, it seems like compactly supported cohomology contains more information than ordinary cohomology. Can one recover ordinary cohomology $H^i_{sing}(X, \mathbb{Z})$ from compactly supported cohomology $H^i_c(X, \mathbb{Z})$? How about if we take coefficents in $\mathbb{Q}$?

I'd love to see "typical", or common examples where various phenomena appears.