I have some intuitions that I want to make precise and accurate. I am very sure there are many mistakes in my understanding, but allow me to state it in the raw form as I sense it for now.
Let $\mathbf{Toph}$ and $\mathbf{CWh}$ denote the category of topological spaces and CW complexes respectively where morphisms are homotopy classes of maps.
Consider the forgetful functor $F:\mathbf{CW}\to\mathbf{Top}$, which induces $F:\mathbf{CWh}\to\mathbf{Toph}$. Suppose for each topological space we fix a choice of CW approximation (basically saying using AC), then the construction of CW approximations yields a functor $P:\mathbf{Top}\to \mathbf{CW}$, which induces $P:\mathbf{Toph}\to \mathbf{CWh}$. Then the following form a pair of adjoint functors $$\mathbf{CWh}\xrightarrow{F}\mathbf{Toph}\xrightarrow{P}\mathbf{CWh}.$$ Moreover, if we consider the induced functors of $F,P$ on the skeleton categories of the corresponding categories, they are equivalence inverses to each other. Therefore $(F,P)$ is in fact an adjoint equivalence.
Now consider the singular chain complex functor $\Sigma: \mathbf{Toph},\mathbf{CWh}\to \mathbf{Com_h(\mathbb{Z})}$, which induces functors on the skeleton categories. By an argument that uses weak homotopy equivalence of CW approximation, it seems that equivalence inverses $(f,g)$ on the skeleton categories are compatible with $\Sigma$ on the skeleton categories. Therefore we can conclude from here that homology and cohomology of CW approximation coincide with those of the approximated space, and as a corollary homology and cohomology are weak homotopy invariant for topological spaces.
Please help me make my statements correct and precise, thank you.
This is all correct, although I don't understand what you feel you're accomplishing with the passing to skeletons. You also shouldn't call the singular chain functor $\Sigma$-that looks like suspension. But, yes, the homotopy category of spaces is equivalent to the homotopy category of CW complexes, and homology and cohomology are well defined on the homotopy category. The composition $\mathbf{CWh}\to \mathbf{Toph}\to \mathbf{Ab}$ of embedding then (say) cohomology is equal to the cohomology functor $\mathbf{CWh}\to \mathbf{Ab}$, so since $\mathbf{CWh}\to \mathbf{Toph}$ is an equivalence its inverse $\mathbf{Toph}\to \mathbf{CWh}\to \mathbf{Ab}$ is equal to the cohomology functors $\mathbf{Toph}\to \mathbf{Ab}$. This can be shown manually as well, as in Hatcher 4.21.
EDIT: There's actually one mistake in your presentation. Your $\mathbf{Toph}$ is not equivalent to $\mathbf{CWh}$, because if $X$ is not of the homotopy type of a CW complex, then $X$ is not isomorphic to its CW replacement, even in $\mathbf{Toph}$. You have to coarsen to $\mathbf{Toph}'$, in which you formally invert the weak equivalences, to get the equivalence you claimed to have shown on skeleta.