Stmt 1: you have proven the reverse direction. You need to show a counter-example where $P(A\mid B) = P(B)$ but $A, B$ are dependent. Instead your only showed an example where $A, B$ independent but $P(A\mid B) \neq P(B)$. It's the wrong direction.
The correct counter-example is not difficult. You can easily construct examples with coins, dice (or even just purely abstractly).
Stmt 2: This is more subtle. $P(A\cap B\mid C) = P(A\mid C) P(B \mid C)$ means that $A,B$ are independent when conditioned on $C$. The question is does that make $A,B$ independent when unconditioned? I would suggest using a 6-sided die and assigning various subsets of results as $A,B,C$ and see if you can first find an example where $P(A\cap B\mid C) = P(A\mid C) P(B \mid C)$.
Hope this helps. If you're still stuck after a while lemme know your progress and I'll see if I can think of further hints.