What is the point with such $z$ coordinate? Either you're missing $\sqrt{\frac{1}{1+(\frac{d\gamma_1}{dx})^2}}$ in $z_{cm}$, or you are adding incorrectly the time derivative of $\gamma_3(\gamma_2(x)$ if you're working with the contact point. In addition, for the second case, the tensor of inertia $I$ would have to be computed with respect to the contact point, $$I=I_ {cm}+\frac{1}{2}Mr(x)^2$$ thus adding a component depending on $r(x)$ (that is why I mentioned the Huygens-Steiner theorem).

Yes. You're right. But I don't think It changes the solvability of the system. In both lagrangian $x$ disappears,in favour of_generalized_ $l$ and $s$.

I've edited my question with your suggestion, and I developed it further. Thanks!

I don't see why the question has been tagged as "homework-and-exercise". I haven't been able to find it on any book, any exercise list. Isn't that to be discussed as a question about classical mechanics, rather than a question on how to apply it? To me, it seems a question about the boundaries of standard concepts in classical mechanics (rolling and slipping).

Any comment would be helpful,including alternative ways to solve (i.e. force detailed analysis, hamiltonian formalism, etc.)