I wanted to get back to this post because I thought it was an interesting way to look at things.
Can the GM cause the pirate crew to plan a mutiny due to the creation of a GM NPC pirate with a backstory and ambitions?
I don't see anything restricting this in a living world sandbox. NPCs, their backstories, and ambitions are the domain of the GM, so they can create what they'd like in that regard.
What I would do here if this was my game, is I'd signal the NPC in question as problematic in some way. That he clearly is outspoken and perhaps even hostile... but I'd temper that by making his role on the ship an essential one. Something that only he can do, or which he is the clear best at.
Then, before it gets to the point of mutiny, I'd let the players interact with that NPC to see if they can change that trajectory. If they can't, or if their actions make it worse, then the mutiny would take place.
Handling it in this way means I've made the situation known to the players, and left it up to them to decide how they want to handle it. I'm not hiding the situation from them. I'm not interested in if they discover the issue... I'm interested in how they handle the situation.
Can the GM cause the pirate crew to be hunted by a sea captain due to the creation of a GM NPC who has ambitions to become admiral, in order to be worthy to marry one of the duke's daughters? Is this not an example of a Living World independent of the PCs?
I would expect that this is absolutely possible in a living world sandbox. It interacts with player choice in the way folks are advocating... like, if the players decide to become pirates, this guy is something they're likely going to have to deal with.
How this is communicated to the players is the question I'd have. I would make it known to them in some way. Like if the game looks like it's going toward this idea of piracy, I'd start introducing relevant elements then, at least conceptually. I'd make sure that in the port city, they heard about this ambitious pirate hunter, and likely a bunch of other possible threats or opportunities as well.
Can the GM cause the mercenaries to be elevated into positions of power within the realm as certain key influentials are targeted and assassinated?
I'm not sure I follow this example entirely. By "the mercenaries" do you mean the PCs?
Can the GM narrate an earthquake which then releases lethal toxins into the air or some other danger from deep within its bowels that hurt this new nation that PCs formed in the hinterlands?
I don't see why not. This could be something the GM prepped as a possibility... perhaps a threat with a countdown of some sort and now it's reached that point.
Just doing so on a whim seems potentially contradictory to what some say here... but I'm not sure how problematic it is as long as it's communicated to the players in some way.
I had a natural disaster happen in a sandbox D&D campaign I ran a while back... a flood struck the twin towns that the PCs were based in (the towns were on opposite sides of a large lake). The flood nearly wiped out one of the two towns (the closer of the two), and severely damaged the other. This was caused by the machinations of one of the factions in play, and then it changed the geography of play, and wound up introducing a new faction entirely.
As GM, I introduced this event and then watched to see what the players did as a result.
Are these reactive or is this the Setting (the GM)?
Almost everything you mention above is the GM. The idea that the GM can't introduce NPCs with agendas that matter to play or threats that exist in the world or events that happen would, to me, go against the idea of a living world.