$K\subseteq X$ compact, C closed, can I say something about $K \cap C$? If C is closed then C is compact as well, but the intersection of two compact set is not compact in general. It would be nice to have this intersection to be compact though, any idea if this is true?

$K \cap C$ is closed in $K$, by the definition of subspace topology, so compact. A closed subset of a compact space is always compact, no extra assumptions needed.
$K \cap C$ need not be closed if $K$ is not closed. 
