Oriented finite CW complex with prescribed homology? 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.
 A: To answer the question of your side note. CW-complexes are not enough, there are needs to work with "convenient categories of topological spaces" where you have all CW-complexes, function spaces, you also want your category to be cartesian closed... This is very important when you want to study iterated loop spaces (see Peter May "Geometry of iterated loop spaces"), bar constructions, classifying spaces...
You can also have at a look at:
Norman Steenrod, "A convenient category of topological spaces", Michigan Math. J. 14 (1967) 133–152
or 
Neil Strickland, The Category of CGWH Spaces. available on his homepage.
The introduction of N. Steenrod's paper is very good and the paper is worth reading. 
Ronnie Brown has done some important works on these "convenient categories":
Ronnie Brown, Some problems of algebraic topology: a study of function spaces, function complexes, and FD-complexes, DPhil thesis (part A), Oxford University, 1961
Ronnie Brown, Ten topologies for X×Y, Quart. J.Math. (2) 14 (1963), 303–319.
Ronnie Brown, Function spaces and product topologies, Quart. J. Math. (2) 15 (1964), 238–250.
