Why does Daniel choose the Crane Stance and kick in the final round? As has already been noted, the move is obvious and nigh-impossible to execute. Simple: Due to the "Sweep the leg" thing, Daniel's left foot can no longer support his weight, but rather than throw in the towel, he gambles everything on a desperation move, the one move he actually has a chance of performing. And it works.
At first, Mr. Miyagi calling Daniel "Daniel-san" given their age difference and relationship seems like improper use of Japanese Honorifics, but then you realize Mr. Miyagi is probably just showing the respect he has for Daniel.
Miyagi is the janitor of the condo Daniel lives in. In a way, he works for him.
In traditional Okinawan karate, only family members were taught- it was literally passed from father to son. So Mr. Miyagi sees Daniel as the son he might have had, and his father saw Sato as a second son to him.
The reason Sato did a Heel-Face Turn. Ya sure the practical reason was that Miyagi saved his life despite being all antagonistic towards him. However Fridge Brilliance suggests that it goes deeper than that. He also reclaimed his lost honor by helping Miyagi and Daniel save the bell girl. In other words he honored a life debt which freed him from his decades long disgrace.
In the second film, Miyagi doesn't want Daniel to come along, giving such reasons as the high cost. Then they reach Okinawa and the fight with Sato flares up again. Perhaps Miyagi's biggest reason was not wanting Daniel to get caught in the middle of it.
In the 2010 The Karate Kid remake, Cheng was particularly jealous of Dre and Mei Ying's relationship. Keep in mind of China's one child policy and how there is a shortage of girls in China. Mei Ying might be the only girl that Cheng had a chance to marry.
Also the scene where Dre falls asleep on Mr Han's lap and his following hesitation to put his hand on Dre's shoulder made a lot more sense when we find out about Mr Han's deceased son who would have been around the same age as Dre.
In their first confrontation, Cheng gave Dre a black left eye. Dre's last blow the won the match landed over Cheng's left eye.
Fridge Horror: What would have happened if, instead of Mr. Han, Dre's mom had tried to intervene when the other kids were beating him up? They probably would have murdered them both.
Fridge Brilliance/Fridge Hilarity: Dre asks Mei Ying to make a pinky swear. Notice that she looked offset when he holds out his pinky finger. It's because that's how you flip someone off in China.
This also gives Mei Ying's parents further reason to be pissed at Dre when he shows his pinky finger at the audition.