$1.$ In the main, how do you determine the boundary curve? I referenced How to find the boundary curve of a surface, like the Möbius strip?, but the answer thereunder refers to the specific example of a Mobius strip.
$2.$ In this question, why is it not $x^2 + y^2 = 1 $ on the xy-plane z = 0?
To answer 2, if you look at how $S$ is defined, it's bounded by the plane where $y = 0$, the $xz$-plane. The equation for the boundary curve follows by simply setting $y$ equal to $0$. You can check that this is the boundary curve by noting that any point on $S$ away from this curve has a neighborhood on $S$ that projects to an open neighborhood in the $xz$-plane.
