Does the above Diophantine equation have other integer solutions besides $(x,y)=(1,2)$ and $(x, y) = (0, -1)$?
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Theorem 6.4.30 in Cohen's Number Theory: Volume I asserts: For each nonzero integer $d$, there is at most one pair of integers $(X,Y)$ with $Y\ne 0$ and $X^3+dY^3=1$. Apply this with $X=-y$, $Y=x$, and $d=9$, to see that that there are no more solutions. Theorem 6.4.30 is attributed to Skolem.
Yes. Here is another one: $x=0, y=-1.$