The following is probably well known, but I wasn't able to locate a reference in the literature.

Let $f$ be a Morse function on a smooth compact manifold $M$ without boundary and let $\rho$ be a Riemannian metric on $M$. As explained in Milnor's Morse theory and many other sources, starting from $f$ and $\rho$ we can construct a CW-complex $M'$ homotopy equivalent to $M$. However, it seems natural to ask whether $f$ gives a CW-structure on $M$ itself, say, such that the corresponding cellular chain complex is isomorphic to the cellular chain complex of $M'$. Is there a reference for that (preferably, one that contains detailed proofs)?

For a generic choice of the couple $(\rho,f)$ one can construct a chain complex (which I believe is called the Morse complex and) which computes the homology of $M$. What is the standard reference for that? This is implicitly done in Milnor's h-cobordism book, chapter 7. Is it true that the Morse complex is isomorphic to the cellular chain complex of $M'$ from question 1?

upd: the original version of the posting contained some very wrong claims and had to be rewritten.

upd1: restored part of question 2 from the original posting. I deleted it thinking it would be trivial, but it seems that it isn't.