$W(x,z)$ is just some function of $x$ and $z$ that gives this set of equations. You use the second equation to eliminate $z$,it can be rewritten as $\partial_z W(x,z) = 0 $, but the form I've given is more convenient. $y$ is involved only in the first equation, you can say that the first equation defines $y$, you cannot get it otherwise.

Indeed, you are correct on this $(y-1)$ factor in the A-polynomial, it is always present in the classical A-polynomial but not in its variations like extremal A polynomials or super-A-polynomial. If I have this singularity, is it still possible to have a CY manifold ( what is meant by CY manifold then? ) ? What changes? There are some examples of CY manifold with singularities like a conifold, maybe something like this?