List all the elements of order 3 in the group $\mathbb{Z_{18}}$ Task:  

Consider the group $\mathbb{Z_{\large18}}$ under the operation of addition modulo $18.\;$
  List all the elements of order $3.$

My professor said the answer was $6$ and $12$.
But isn't the answer $0,6,12\,?$
Because
$$\begin{align} \langle 6\rangle = & 6,\\ &6+6 = 12,\\&6+6+6 = 0\end{align}$$
Or is it not necessary to include $0\,?$

You've identified the elements in the subgroup of order three:  $$\{0, 6, 12\}$$
You were asked to find the elements of order $3$. Only two of those elements in the subgroup have order $3$ (each of $6$ and $12$ generates the subgroup above). Indeed, $0$ is the identity element of $\mathbb Z_{18}$, and also of the subgroup above, and has order one.
