The elephants used in the early Weissmuller films were not African but Asian elephants with African sized ears strapped on. This practice is still common because the Asian species is much more docile than the African. In later films the fake ears were abandoned altogether figuring no one would know the difference.
Tarzan's distinctive call was either created by sound recordist Douglas Shearer from various sounds, or it was indeed Johnny Weissmuller doing the yell himself. Co-star Maureen O'Sullivan insisted throughout her life that it was Weissmuller doing the yell without any technical assistance. Johnny Weissmuller explained on the Mike Douglas show that he was the one that did the yell, which he did on the show.
Some of the monkeys used in this film got away, and they are still breeding in the wilderness of Silver Springs (in Ocala, Florida).
Apparently, the chimpanzees were supposed to be the baby apes and actors in ape costumes were the adults, a practice that was dropped after this film (one of the men who played the apes was Ray Corrigan). The same set up was used over 30 years later in 2001: A Space Odyssey (1968), with chimps playing the children of the costumed primate group.
In a television interview, Johnny Weissmuller said that he initially tested for the Tarzan role as a way of seeing friend Clark Gable. After running, jumping, and swinging on a vine, he was offered the part, but he refused to change his name when a studio exec suggested it. He had never heard of Weissmuller. A second exec told him Johnny was an Olympic swimming champion, so he suggested they could "put some swimming in it". Thus, one of Tarzan's primary traits was added as an afterthought by a studio head who didn't know he had the world's greatest swimmer as his star.