Morse theory gives you a cell decomposition for a manifold; to go from this to something like a triangulation is a purely topological problem. It is quite often non-trivial, however.

I think what you say is almost true (and follows from Yoneda's lemma) if you replace "abelian" by "finitely presented". Should not be too hard to prove, in fact. And for abelian there should be an easy counterexample. Sorry for being not very specific.