Complex numbers question

Taking the modulus gives $|z|^2=|i\overline{z}|=|z|$ hence $|z|=1$ since $z\not=0$. Multiplying both sides of the original equation by $z$ then gives: $$z^3=i\overline{z}z=i|z|^2=i$$ that is $z^3-i=0$.

If you call the solutions of that $z_1$, $z_2$ and $z_3$, how can you find the numbers $z_1+z_2+z_3$, $z_1z_2z_3$ and $z_1z_2+z_2z_3+z_3z_1$ straight from the equation?

Try writing $z = a + ib$ for $a,b \in \mathbb{R}$ and expand $z^2$ and $iz^-$. Then compare real and imaginary coefficients in the two expansions.