When Daniel asks Miyagi why he spared Kreese, he tells him that "living is worse than death". As Cobra Kai shows, Miyagi-Do does have killing techniques, and Miyagi knows them. He would rather have Kreese suffer a fate worse than death than kill him, let alone teach Daniel them until he was able, though, Chozen is the one who teaches Daniel the stronger paralysis techniques.
The reason Sato did a Heel–Face Turn. 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 Yuna. In other words, he honored a life debt that 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. Yet at the same time, Miyagi is deeply touched by Daniel's desire to come with and zest to learn about his mentor's home and culture. Miyagi knew seeing his dying father, lost love, and old friend-turned-nemesis again would be extremely painful, and was grateful to have his student, his surrogate son, by his side to help him through it. And Daniel wants to be there support his beloved mentor, friend, and father figure.
Sato doesn't directly attack Daniel because he has no quarrel with him. Chozen's actions against Daniel directly are his own doing and not on Sato's orders, and the reason Sato doesn't encourage or interfere is because he's honorable enough to leave the quarrel between Daniel and Chozen. The exception being when he was betting during the ice break challenge, only because Miyagi was betting too.
For years, it seemed that Sato has been trying to break the log that he and Miyagi found when they were young and has been unsuccessful in doing so. During the storm, he is pinned down by a board that is the same weight and size as the log and unable to move, yet Miyagi breaks it and frees him. You then remember the breathing technique Miyagi taught Daniel to help him focus and concentrate. Sato has not been able to break the log all this time because he has been so consumed by anger and rage over what happened that he cannot concentrate on anything else whilst Miyagi has let go of the past and that is what enabled him to free Sato.
There's also the fact Sato assumed Miyagi was a Dirty Coward for leaving instead of fighting him to the death. Seeing Miyagi break the log, it becomes clear Mister Miyagi just didn't want to kill his best friend. Seeing the proof Mister Miyagi is far superior at karate, he's left humbled as well as aware of just how close he came to death (or at least humiliation) all those years ago.
There's an assumption of a Translation Convention or just being convenient for the audience that everyone in Okinawa seems to be speak English so Daniel can talk to them. Except, the movie bothers to justify it. Tomi Village next to a US Air Force base (well, the reverse is true) and Chozen says, "It's good business to speak English." They've had about thirty years to learn English if it was constructed right after WW2.
The "drum" technique in Karate Kid 2 is just a roundhouse punch. The thing is, it actually is explained to be the "basis" of karate and this is a blindingly obvious truth if one realizes it. It is an incredibly simple but effective way of hurting someone badly. Also, the drum is a visual metaphor for what one has to do to throw them effectively. Which fits into Miyagi having many ways to teach fighting techniques.
Also, why does everyone in the village have one of the little drum tools? It seems a local tradition that would make sense at fights given it is apparently the village karate originated.
During the ice breaking challenge, when Daniel does his "breathing technique", almost everyone in the bar laughs at him. Except for Sato, who watches Daniel closely. Given that the breathing technique was taught to Miyagi by his father, Sato surely recognized it, since Miyagi's father was his sensei as well. Similarly, he is the next after Miyagi to take out the hand drum, implying he's also familiar with the drum technique.
This connects with Chozen's later appearance in Cobra Kai, where he reveals that there were things Miyagi didn't see necessary to teach Daniel... As this is something Sato didn't teach Chozen. Both Miyagi and Sato had different ideas of what was better to teach to their students and preserve for the future, and acted on it.
More Fridge Brilliance: The breathing exercise teaches patience and self-control. Perhaps if Sato had recognized its value and taught it to Chozen, both would be different characters.
