Finding scalar that creates a point with zero X-coordinate for popular elliptic curves
No (See 2.). An alternative way to test a library could be to change the base point. You can find the point with x coordinate equal to zero, multiply it by a random $r$ and then use the result as the base point of ECDSA with $k=r^{-1}$.
This is equivalent to solving an ECDLP (discrete logarithm problem) which on standard curves, is believed to be hard (practically unfeasible)
This is possible.
$s=k^{-1}(z+rd)$ which is 0 when $(z+rd)=0$. If you can freely choose $z$ (e.g. the library receives it as parameter, and doesn't compute it internally) then you can choose $z$ such that $z=-rd$. Otherwise, you can select $d$ such that $d=-\frac{z}{r}$.

Note that another corner case that I test on implementations of ECDSA is where the $x$ coordinate of $k*G$ is greater than the $G$'s order. This again is done with the same trick I described in point 1.