There are many interesting triangulated categories without dg-models, i.e. the stable homotopy category. There is a new paper by Stefan Schwede (arxiv.org/abs/1201.0899) addressing the question of how and to what extent triangulated categories with such "topological models" can fail to have "algebraic models" in the spirit of dg-categories.

One more thing -- it is not true in general that weak equivalences in a model category satisfy the calculus of fractions. It only becomes true after dividing by the homotopy relation, for which you need to use cylinders or path objects. Think about the classical construction of the derived category of an abelian category. You have to divide by the chain homotopy relation before you can prove that quasi-isomorphisms satisfy the calculus of fractions.

I'm afraid I still don't understand what the question is about and how this answer addresses it. In any model category a map is a weak equivalence if and only if it becomes an isomorphism in the localization. If we start with the standard model structure on $\mathrm{Top}$ with weak homotopy equivalences as weak equivalences, then we don't recover actual homotopy equivalences this way.

This is besides the main point of the question, but are you sure that this works for levelwise cofibrations? It seems to me that in order for the composition to satisfy the pushout product property you need a stricter notion of cofibration. I mean the one where we require that in the naturality squares for cofibrations the induced map from the pushout is a cofibration.

I made a correction. My counterexample is not a homotopy pushout, but not for the reason I originally stated.

My answer is essentially the same as yours. (I didn't try to "improve" it, I'm just a slow typer.)

There is no such functor as your $R$. You cannot just "remove all degenerate simplices" since some degenerate simplices may be faces of non-degenerate ones. You can consider all simplices which are faces of non-degenerate simplices, they form so called "core" of a simplicial set. However, the core is not functorial. Of course both adjoints you mention exist, but they go the opposite direction than your $R$.

@Akhil: The map $\Delta[1] \to \Delta[2]$ induced by $\delta_1 : [1] \to [2]$ is both left and right anodyne, but it's not inner anodyne since it's not bijective on $0$-simplices.