Can$A \cap (B' \cap C')$ be $(A \cap B') \cap (A \cap C')$? If I use the above statement, provided that it is right, in a question, would I have to prove it as well?

In this case, the proof is rather trivial: since the $\cap$ operator is both associative and commutative, all the following are equivalent:
$$ (A \cap B) \cap (A \cap C) $$
$$ A \cap B \cap A \cap C $$
$$ A \cap A \cap B \cap C $$
$$ A \cap B \cap C $$
$$ A \cap (B \cap C) $$
As others said, whether or not you actually have to include the proof is mostly a matter of context.
