Moore spaces are finite CW complexes with prescribed homology, but they may be non-orientable and even not topological manifolds. Are there oriented, connected CW complexes with prescribed homology (with $\mathbb{Z}$-coefficents?) If such a complex is also a topological manifold then there are additional restrictions (Poincare duality) so we can't expect such spaces to be homotopic to topological manifolds.

As a side note: Should I consider CW complexes as the "most general & nice" class of topological space that people commonly work with upto homotopy. (This is really a Yes/No question intended to be interpreted as basic. For example, the class of CW complexes (upto homotopy) includes all other classes of topological spaces I know: topological manifolds (with or without boundary), simplicial complexes, delta complexes. In fact, except the class of topological manifolds (with or without boundary), all of these describe the same class of topological spaces upto homotopy) For me, topological manifolds should have a countable basis.