Dawson Casting: Ralph Macchio was 21 when the first film was made, playing a 17 year old. He was 28 playing an 18 year old in the third film. Not many people noticed thanks to Macchio's boyish good looks and high-pitched voice, making this one of the few effective examples of the trope. Avoided in the reboot, as the kids are actual kids.
Hey, It's That Guy!: The black Cobra Kai student? It's Lamar Latrell. We shit you not. Karate Kid II had one line roles from such up-and-coming actors as SVU's B.D. Wong, and Clarence Gilyard of Walker: Texas Ranger and Matlock fame.
Before Robert G Durant became synonmous with burning scientists and collecting fingers he was a racist drunk that had his beer bottles Karate chopped by Arnold.
Throw It In: The song drunkenly sung by Mr. Miyagi was one he heard when he was a child. This even gets worked into the second movie as being Miyagi and Yukie's old song.
Also, the song playing in the country club Daniel crashes in the first movie is heard again in the second movie, on the radio in the cab ride from the Okinawa airport. Neither occurrence is a plot point.