Given the same triangle, of the statements (i) $\angle ACB = 30^\circ$, (ii) $\tan A = \sqrt{3}$, (iii) $\sin(A + C) = 0$, which are correct?

Here $\angle ACB=30^\circ$ and $\tan A=\tan 60^\circ=\sqrt{3}$ are both true, but $\sin(A+C)=\sin 90^\circ=1\neq 0$, so statement (iii) is false. Hence only (i) and (ii) are correct.