Determining if a context-free grammar produces even-length strings
There is an algorithm:

Recall that given a CFL $L$ and a regular language $K$, the language $L\cap K$ is also CFL (and a CFG for it is computable given a CFG for $L$ and a DFA for $K$).

Now, observe that the language $K=\{w: |w| \text{ is even}\}$ is regular.

From this we conclude that given a CFG $G$, we can construct a CFG for the language $\{w: |w|\text{ is even and }w\in L(G)\}$, and we observe that this language is nonempty iff $G$ generates a word of even length.

Since checking emptiness of CFG can be done in polynomial time, we can solve the problem in polynomial time.