Prove that $I(A;B|C)=0$ given $I(A;B)=0$
Your claim is false. Let $A,B,C$ range uniformly over the set
$$
\{(x,y,z) \in \{0,1\}^3 : x \oplus y \oplus z = 0 \}.
$$
On the one hand, $I(A;B) = 0$. On the other hand, $I(A;B|C) = 1$.