* First, I build the PCs - Thurgon, a knight of a holy military order (the Knights of the Iron Tower), and his sorcerer sidekick Aramina. The GM tells me that we're starting play on the Pomarj-Ulek border - that's a bit warmer than I had expected (in my initial conception Thurgon is rather Germanic) but I roll with it. The backstory I've written for Thurgon includes that "Thurgon left the Iron Tower only weeks ago. The Knight Commander of the order sent him forth into the wilderness. He does not know why." And also that Thurgon has not set foot there in Auxol, his ancestral estate, for over 5 years, since he left to take service with the Iron Tower.
Let's explore how a very similar story emerges in living sandbox play.
All of this you listed above is backstory and is applicable to living sandbox play as well (only caveat is that the GM would have some power to veto particular background elements). There's no divergence yet assuming the background was accepted.
* Now there are some ambiguities in Thurgon's background as represented by some build elements: there is an Affiliation with the Order of the Iron Tower; and also a reputation as The Last Knight of the Iron Tower. So it's not clear if the Tower has fallen, or is falling. The GM doesn't push for certainty in that respect.
When it comes to whether the Iron Tower Order has fallen the GM would have some notes regarding them and their enemies. He would then proceed to use a non-player facing mechanic to determine what happened to them. The players wouldn't necessarily learn of this happening unless they went out of their way to explore it.
Instead, he starts fairly low-key and as one might expect: we (that is, Aramina and Thurgon) are travelling along the river frontier (between the settled lands of Ulek and the wilder lands of the forest and the Pomarj), where there are old forts of the order (now abandoned) and also abandoned settlements.
No issue kicking off a D&D sandbox campaign in such a manner - especially one with so much backstory.
* At one of the homestead, I declared a couple of checks: a Homestead-wise check (untrained) to learn more about the circumstances of abandonment of this particular ruined homestead, which succeeded, and hence (in this case) extracted some more narration of backstory from the GM; and then a Scavenging check, looking for the gold that the homesteaders would have left behind in their panic and which the orcs would have been too lazy to find.
D&D Sandbox players wouldn't necessarily declare their own checks, but instead would say something like I investigate the area for clues about what caused this homestead to be abandoned. I also check the obvious spots for where there might be gold. Drawers, underneath mattresses, chests, etc.
Nearly everything so far fits nicely into a D&D sandbox campaign.
Unfortunately this second check failed, which meant that Orcs from a raiding party had virtually infiltrated the homestead before I noticed them. Here we have an attempt at a player-authored plot moment, but the failure tilts the balance of narrational and hence situational authority back to the GM. The fight with the Orcs engaged Beliefs and Instincts, so there were local moments that expressed Thurgon's character in this bigger GM-established context.
And now we have our first major difference with typical D&D sandbox play and story now. Your check for gold determined the presence of the orcs. In a living world sandbox the most likely reason for encountering the orcs might be the PC's lingering to long triggering a behind the scenes GM check to see if the orcs arrived. Most likely the orcs engage the PC's in combat.
* The Orcs (as the GM narrated things) were part of a larger raiding party, with mumakil. I think the GM was hoping I might chase the mumakil, but I have no animal handling, animal lore etc and so the mumakil remained nothing but mere colour.
Sounds very much like typical D&D living sandbox play. Players can choose to engage or not engage with elements presented.
The larger raiding party was chased off by a force of Elves, again narrated by the GM.
Very possible in sandbox play (probably based on triggering event) instead of however the Elves were generated in the Story now fiction.
I wasn't surprised that Elves should show up - my GM loves Elves! I tried an untrained Heraldry check to recognise the Elves' arms, and failed - so the Elven leader was not too taken by me! In this there was cross-narration by me and the GM, but it ran in the same direction: as I was saying (in character) that I don't recognise the Elven leader's arms and wondered who he was, he (spoken by the GM) was telling me that he didn't like my somewhat discourteous look.
For the same action a D&D GM would probably call for a history check and the playthrough afterwards could be very similar.
I don't know what, if anything, the GM had in mind for the Elves, but one of Thurgon's Beliefs was (at that time) that fame and infamy shall no longer befall my ancestral estate. So I invited the Elf to travel with his soldiers south to Auxol, where we might host them. The GM had the Elf try and blow me off, but I was serious about this and so called for a Duel of Wits.
Seems very possible for sandbox play (though sandbox play doesn't require such in depth back grounds, though it isn't hostile to them either) just replace duel of wits with persuasion and the same thing happens in living sandbox D&D.
Unfortunately my dice pool was very weak compared to the Elf's (6 Will dice being used for untrained Persuasion, so slightly weaker than 3 Persuasion dice vs 7 Will dice and 6 Persuasion dice) and so despite my attempt as a player to do some clever scripting I was rebuffed by the Elf without getting even a compromise. Here we have a player-authored plot moment. Although it ended in failure for the PC, it was all about what I as a player had brought into the situation. I'm pretty sure the GM hadn't anticipated this. So I don't know what he anticipated for the Elves' departure, but in the game it followed my failure to persuade them to join me.
Given how closely the living sandbox play mimics this I don't see why the same things aren't being said of it.
And to play devil's advocate a moment - who brought the orcs in, who brought the elves in, the whole situation wasn't about what you as a player brought into the situation - though a good portion of it was?