All solutions here are SUGGESTED. Mr. Teng will hold no liability for any errors. Comments are entirely personal opinions.

(i)
$z^3 = (1 + ic )^3$

$= 1 + 3ic - 3c^2 - ic^3$

$= 1 - 3c^2 + i(3c - c^3)$

(ii)
Given $z^3$ is real,

$\Rightarrow 3c - c^3 = 0$

$c = 0 \mathrm{~or~} c^2 = 3$

$\therefore, ~ c = \pm \sqrt{3}$ since c is non-zero.

$z = 1 \pm i \sqrt{3}$

(iii)
$|z| = 2$

$|z^n| = |z|^n = 2^n$,

$2^n > 1000$

$n > 9.96$

Least n = 10.

$arg(z^10) = 10 arg(z) = \frac{2}{3} \pi$