In the "Omega Glory" the Yangs are taking back their 'own' lands. The Comms have had them oppressed for centuries. It was Tracey who interfered first by attempting to change the balance of power by arming the Comms. Kirk accidentally worsened the situation with his men's phasers.
Still in the end the Yangs regained power in their own lands despite Tracey's interference.
I don't think Kirk violated the PD here, maybe it was Spock by arranging the communication to Sulu and letting the natives see a beam down. Telling them to read their own document would be a very soft way of breaking the PD IMO.
I'd have to know more about "The Apple" to judge. Were those people we saw on screen the only ones on the planet? Did they build the thing? Did the machine capture them? Do they have free will? I'm not so OK with sentient beings being the slaves of machines (call me racist if you want to)
In 'Miri' or any other planet they should check for bacteria etc first but that would slow the story down a lot. On any new planet in reality, they would probably been someone down in a decontaminated Hazmat suit, collect samples and examine then. If deemed safe a team would then beam down after being decontaminated so as not to pass Federation diseases onto each new planet.
They also should not allow any random person they let on the ship to have free reign of the ship but in every Star Trek they do - just to have some drama, conflict, a story.
Would Picard have left the kids? I hope not, though technically I think he might feel compelled to. He wasn't moved by the little girl in 'Pen Pals'... If I were Picard I would just view this as a bunch of human children who needed rescue. They didn't have a 'civilisation' to not interfere with.
Even Kirk's action of leaving a team there seemed strange. The kids should have just been transferred to Earth to be adopted or placed in reform school.
(Teenagers saddled by small kids aren't really going to be able to rebuild a civilisation) They really had nothing to keep them on the planet.