Question: Since World War 3 was the cause of the nuclear devastation, as evidenced in Beneath the Planet of the Apes, how is it that Dr. Zaius doesn't know the full story of the rise of the apes over the humans? Wouldn't the story be passed down?
Answer: Dr. Zaius did know the true history of man and ape, but he deliberately hid the truth from the other apes. For Zaius (and other high-ranking apes who were guarding the secret), it would be shameful and demoralizing to ever admit that humans were far superior to apes in the past and that they could, potentially, conquer the apes. In more than one scene (such as the paper airplane scene in the first film), we see Zaius obviously frustrated that Taylor's very existence threatens to expose the truth.
Question: Was Doctor Zaius really aware of what happened to the human race? Many times during the film, he does things that indicate such a possibility, like crushing the paper airplane.
Answer: Yes, he knows what happened and does whatever he can to suppress knowledge about it. He does not want his kind to know that humans were once far superior, which he believes led to their destruction.
Question: World War 3 may have been the reason for the nuclear war, but what caused World War 3?
Answer: It's never stated what caused it (in this movie, at least). So much time has passed that historical records have been lost or destroyed, and the remaining humans are no longer literate, nor do they have any inkling about their species' past. The apes also do not appear to know the reason, only that mankind somehow destroyed their own civilization.