Lower-bounding the Membership Problem in the Bitprobe Model
First, the reason why the authors had to omit so many details is that the paper was published in a conference (SOCG) with a strict page limit. When you publish your paper, please provide a full version on your website with all the proofs. In my area usually the full details are in an appendix which is not available from the conference website since back in the day, the proceedings were printed and so a strict page limit had to be enforced. Nowadays many proceedings are online only and then there is no page limit, but it seems that your paper dates from before that period.

I'll only address your main question. Consider the structure for storing $\{x,y\}$. When you query $x$, you get either "left" or "right" at the first probe. Say you got "right". Then the right endpoint of the edge $x$ must be 1 (we use the convention that the output is the last bit read, see the paper). When querying $x'$, we read the same first bit, and so the right endpoint of the edge $x'$ must be 0. This means that the left endpoint of $y$ must be 1. This means that when querying $w'$, the first bit read must be "right". But then when we query $w$, we read the right endpoint of the edge $w$ which is 1.

Suppose now that you got "left" in the first probe of $x$. Then the left endpoint must be 1, and so the first probe of $y'$ must be "right". Therefore the right endpoint of $y$ must be 1, and so the first probe of $z'$ must be "left". But then when we query $z$, we answer the left endpoint of $z$ which is 1. In both cases we have reached a contradiction.