Han and Li fought after the tournament.
Similar to the original movie, as the Parkers, Mei Ying's family and Han are leaving the tournament, they see Li viciously berating Cheng for his second place finish. Li destroys Cheng's trophy and kicks him off the sparring team. When Cheng protests, Li puts him in a chokehold and fends off the other Cobra Kais. Han orders Li to release Cheng, but Li refuses. Han easily frees Cheng, angering Li, who lunges at Han. Han's quickness soon has Li at a disadavantage. When it appears that Han is about to deliver a fatal strike to Li who urges him to do so, Han simply refuses and lets him go telling him he's not worthy. The group depart as Cheng and his classmates quit the Li's dojo en masse.
- It happens in the alternate ending, although the instructor was about to attack Cheng instead, Han stops him and fights Li. At the end, Han is about to use a fatal strike, but was stopped by Dre. Instead, the mother basically bitchslaps Li. Yes, the mother bitchslapped one of the instructors.
Ali is the real villain of the first movie
Okay hear me out. Yes Kreese is a huge douchebag, but he never actively encourages his students to pick on those weaker than themselves. He is only shown as being merciless when it comes to fighting, and given where he received his training, the military most likely during Vietnam, that makes sense as anyone who has learned MCMAP or Krav Maga would tell you. In his dealings with Mr. Miyagi coming to his dojo and accusing his students of picking on Danny, Mr. Miyagi has insulted his honor as their sensei, by accusing him of teaching them improperly. This would be an insult in any real dojo. However, if we look at Ali throughout the movie we notice that she really is the reason for why Danny and Johnny are fighting and at no point does she try to get them to stop. She actively hangs out/flirts with both of them, and the same guys that Johnny hangs out with and beat up Danny are also her friends and at no point is she ever shown refusing to hang out them or trying to get them to lay off of him. Instead she insists that he learn to fight that way he can beat Johnny up. This troper has come to the conclusion that Ali gets off on watching boys fight over her so she tries to facilitate this as much as possible. Johnny is hopelessly in love with her and she keeps him on an emotional leash by constantly breaking up with him, not talking to him, getting back together with him, flirting with other guys and generally making things confusing for him. Johnny just wants to be with Ali, Danny is not used to getting much attention from girls and whenever Danny does try to leave Ali alone as she keeps getting his ass kicked, she actively seeks him out in public, at school where Johnny or his friends can see, to get him beat up again because she likes to watch it.
- Goes hand-in-hand with how she drops him on his ass in the second movie...
Johnny and Bobby quit Kreese's dojo and become friends with Danny
Ok, I have yet to see the second movie so don't know if it's mentioned that they do quiet there, but giving Bobby begged for Danny's forgiveness after the attack on his knee and Johnny saying Danny was alright and being gracious in defeat it seems the natural things to do that they'd quiet the dojo that ordered them to cheat and become friends with Danny.
- I like to think that Bobby did at least anyway. Can't see it happening with Johnny, who'll probably always hold a grudging respect for Daniel but nothing more. But Bobby? Becomes his best friend and names his first son Daniel (In my head canon anyway.).
Daniel and Bobby eventually start their own karate school together
They become business partners/instructors in the Kesuke Miyagi Karate Dojo and teach martial arts to a new generation of Karate Kids.
Daniel and Dre will meet each other in the Sequel of the New Karate Kid.
Much like in the Star Trek
movie when original Spock meets movie Spock. Daniel and Dre will meet in some point where he does a Passing the Torch
Miyagi taught Dre's Grandfather.
First of all Dre claimed he learned Karate from his grandfather. Dre and his mother are from Detroit and Miyagi got his cars from Detroit as well. While Miyagi-style Karate has been passed down from Father to Son, Miyagi has has made exceptions more than once, with Julie's grandfather, Daniel, Julie and even asked his father to teach Sato.
At the end of Part 3, either Miyagi or Jessica had a surprise waiting...
...in the form of the police, there to arrest Bad Boy and his minions for the cliffside extortion. They in turn dropped the dime on Kreese and Silver so fast, the psycho pair tried something, which turned into an assault on a police officer. Daniel sued and got Silver's wealth, using it to clean up the mess he made, hence his and Miyagi parting ways.
Julie's Grandmother isn't alone in Miyagi's California house.
Miyagi asked Daniel to keep an eye on her while he helps Julie out.