Is the set of CFGs that contain all odd and even length words Turing-decidable?
The problem is that you interpret e.g. "grammars that generate all even length strings" as "grammars that generate only all even length strings", which is wrong. If you look at it this way, the intersection you are looking for is grammars that generate all even length strings and all odd length strings, i.e., grammars that generate all of $\Sigma^*$