The music that Miyagi turns on when he is teaching Julie to dance is the same song that Johnny and Ali dance to at the country club in the original The Karate Kid (1984) and is playing on the radio in the car that picks up Daniel and Mr. Miyagi in The Karate Kid, Part II (1986).
The Kata Hilary Swank performs in the monk's hall and under the waterfall appears to be a portion of the first kata (form), "Naihanchi Shodan". This kata, historically of Chinese origin, is common in Okinawan and Japanese karate as well as Tang Soo Do, (a Korean art) practiced by the film's martial arts choreographer, Pat E. Johnson. Miyagi-San identifies his style only as "Okinawan Karate", but that could be any of the Okinawan styles that share this form. Since in The Karate Kid, Part II (1986) Miyagi's nemesis runs a school in Naha with the Goju Ryu fist as a logo and his students wear the Goju Ryu fist on their uniforms, then Miyagi is possibly teaching Goju Ryu (one of the five main styles of Okinawan karate), since Miyagi's father taught both him and Sato.
The Karate Kid (1984) made $90.8 million, The Karate Kid, Part II (1986) grossed $115.1 million and The Karate Kid, Part III (1989) earned a disappointing $38.9 million. Nevertheless the studio was keen to continue the franchise, though its $8.9 million take effectively killed the prospects of any further films. After a sufficient lapse of years, the 2010 remake took $171.8 million at the box office.
This is the only film in the series in which Mr. Miyagi does not appear in the final frame of the movie before the credits begin. However, he does appear in the last frame of the scene that transitions into the scene the credits play over.