Say we have a vector in $\mathbb Z^3_5$: $v= (1,2,0)$ it looks like it isn't the zero vector but if we multiply it by a scalar: $5v=(5,10,0)\overset{mod5}=(0,0,0)$ so now it is the zero vector and we only multiplied it by a scalar. This doesn't happen in other vector spaces.
My question is, given the above example, how can I know whether or not a vector is the zero vector in a modulo space? When am I allowed to reduce it like in the above example?
$(1, 2, 0)$ is not the zero vector in $\mathbb{Z}^3_5$. A vector $(v_1, v_2, v_3) \in \mathbb{Z}^3_5$ is zero if and only if $v_1 \equiv v_2 \equiv v_3 \equiv 0 \mod{5}$.
When you multiply a vector in $\mathbb{Z}^3_5$ by a scalar, you should realize that sinze $\mathbb{Z}_5$ is the base field of this vector space, any scalar should be thought of as an element of $\mathbb{Z}_5$. Thus, you can see that the scalar $5$ that you multiplied by is equal to $0$ (in the base field)! And of course, multiplying any vector by $0$ yields the zero vector, regardless of whether the original vector was zero or not.