Is $Y\cap(X-C) = Y\cap{X} - Y\cap{C}$? Is $Y\cap({X-C}) = (Y\cap X) - (Y\cap C)$? I tries using the definitions of intersections and difference of sets but am not able to prove it.

Yes:
\begin{align}
(Y \cap X) - (Y \cap C) &= (Y \cap X) \cap (Y \cap C)^C &\text{ (By definition of difference)}\\
&= (Y \cap X) \cap (Y^C \cup C^C) &\text{ (By de Morgan)}\\
&= (Y \cap X \cap Y^C) \cup (Y \cap X \cap C^C) &\text{ (By distributive law)}\\
&= \emptyset \cup   (Y \cap X \cap C^C) \\
&= (Y \cap (X \cap C^C)) &\text{ (By associativity of intersections)}\\
&= Y \cap (X-C)
\end{align}
