Context free language and the complement of it
Sorry, your approach does not work. 

Breaking down $L_1$ into two parts is a good idea. Unfortunately you proof these parts are context-free cannot be correct, as one can give a grammar for each of them.

As for the complement, there is no such thing as a intersection rule. The fact that context-free languages are not closed under intersection simply states that there exist two context-free languages the intersection of which is not context-free. The intersection of any random pair of context-free languages can be context-free (e.g., if the intersection is empty).

Back to the drawing board.