If an event $A$ is independent from $B$ and $C$, is A also independent from $B \cap C$? $A$ and $B$ are independent. $A$ and $C$ are independent. Are $A$ and $B \cap C$ also independent?

No. Pick two numbers $x,y$ from $\{0,1\}$ independently and with prob 1/2 each. Let $B$ be the event that $x$ is even, $C$ be the event that $y$ is even and $A$ be the event that $x+y$ is even.
