Why does strictly Walrasian demand with quasi-concave utility function mean that the walrasian demand having only one single consumption bundle?
For case 1, you can argue $z$ will be unique by contradiction:

Suppose ad absurdum there is another $z'$ that is feasible (i.e. $p^Tz' =w$) optimal and $z'\neq z$. Then you can consider a convex combination of $z$ and $z'$: $\bar z = \beta z + (1-\beta) z'$, for $\beta\in(0,1)$. Notice that $\bar z$ is still feasible (because it is a combination of two feasible bundles) and, because of strict quasi-concavity, $u(\bar z) &gt; u(z')$ so that $z'$ cannot be optimal, a contradiction.