The first is true in any space by the definition of the subspace topology: every set $U$ open-in-$C$ is of the form $U'\cap C$ for some set $U'$ open-in-$X$.

Similarly, in the second statement $A$ and $B$ are already closed-in-$X$, so $A$ and $B$ would be be their own extensions, and $U'$ and $V'$ exist by normality of $X$.