Not a sandbox, as the PCs are not really free to explore the world. I agree with Pawsplay that sandbox implies exploration as a focus of play, and that is not the case in my game.
You keep subconsciously thinking of exploration in terms of physical spaces as if that was the only or even most interesting sort of exploration.
I went and tracked down the posts you refered to about your prep and play style, and you are firmly in the sandbox style. There are lots of clues, but to me the most telling was this quote:
To keep the momentum going, in my experience you need (i) a fairly rich and layered myth/history, with (ii) lots of interrelated elements that both relate to the PCs' backgrounds (and players' thematic concerns) and will give the players lots of scope to make choices about alliances/enemies/how to engage, and (iii) a story structure (established via geography, antagonist's motivations, etc) that makes it plausible for multiple such choices to be made and have their ramifications play out over many sessions of play without things being forced to an early conclusion (the game can become derailed if things go too far in this direction, and the story is so convoluted, dense or just plain slow that no progress towards a conclusion seems to be possible). And obviously the players have to be happy to buy into this.
The upshot is fairly complex plots with the PCs at the centre of events in the unfolding history of the gameworld. The game can become fairly sprawling, in terms of the relevant geography and the PC's salient field of action, but it's not a sandbox (ie the players aren't exploring the world with their PCs - if anything, they are exploring their PCs with the world as a tool in that endeavour). At the practical level, players ensure at least some of the PCs have sufficient knowledge skills to engage with the gameworld, and they take lots of notes to keep track of everything, and draw relationship charts or similar to keep track of enemies/allies/factions/historical connections etc.
All of that speaks to sandboxing. In particular, I like the last bolded session because it tangibly describes the real geography that is being primarily explored here (though it seems alot of physical geography is being explored as well). If the players are drawing relationship charts to keep track of the many enemies, allies, and factions they are relating to you can lay money that you are in a sandbox.
The fact that you've worked out a complex mythology for your game world and you put your characters into it seems also to me to be very telling.
All in all I'd describe you as mostly sandbox, done in what sounds like a mixture of the High Drama and Low Drama styles. You've got a nice little fantasy soap opera going by the sound of it.
Well, that seems to me to rule out an adventure path as "good GMing", becaus in an adventure path the players are not free to choose who shall be their PCs' enemies and allies (except in some marginal cases). If the PCs decide to join rather than try to overthrow the evil cult, the adventure path grinds to a halt.
Depends on how good the DM is at improvising. A good DM will retool and start up again, creating a new adventure path on the fly beginning at the unexpected fork in the road that destroyed his prior plans, reusing and repurposing whatever material he made in advance while still following the broad outline of his story arc only with the PC's now in the role formerly occupied by antagonists. The 'adventure path' portion of my current campaign is actually fairly resilient to the player's changing sides, and the general theme of a world spanning chase that is my inspiration doesn't change no matter which side they are competing for.
When something like this occurs and the players 'get off the train', you just bring out a new set of tools and conflicts that accord with the goals your players have implicitly stated they are more interested in. And the fact that we now have a branching tree shows that while many groups may go through an AP in a text book linear way, others may end up in some complex gray area which isn't entirely one or the other.
This precludes linearity, because there is no predetermined destination.
Even if I hadn't established from reading your other posts that you were a sandboxer, the very fact that your game precludes linearity to me would be a strong indication that you were.
A brief actual play example: when the PCs encountered the duergar slavers, I had assumed that they would fight them as a prelude to raiding the duergar stronghold to free the prisoners. Instead they negotiated a deal with the slavers to ransom the prisoners for a share of the loot taken from the hobgoblins from whom the duergar had bought the prisoners. (Mechanically, this was resolved as a skill challenge.) So instead of a fight plus a fortress raid, the game is headed towards a settlement of the ransom transaction at a designated time and place.
And, while I think I've established that you are sandbox, from this snippet I couldn't really tell. To provide you with an example, when I first began plotting out my campaign there was one particular character that I had intended to bring into relationship with the characters in the role of an advisor/mentor figure. However, because of a series of events that I did not foresee, the players have gotten themselves into a quite hostile relationship with the character so that in his future role in the story he is more likely to be a foil or even antagonist than any sort of ally. Likewise, in any reasonably good adventure path, the players would still have the oppurtunity to resolve a conflict with the duegar slavers in any number of ways - combat, stealthy raiding, bribes, diplomacy, winning an ally willing to initiate military conflict, etc. The way I know that you are likely on a sandbox isn't because the players had freedom to choose how to resolve a conflict, but because I suspect they had some control over the conflict they'd choose to resolve and some say over what conflict they'll address after the ransom transaction is complete. Moreover they have apparantly complete freedom over 'where they go' in the terrain you've created, which is 'political' for a lack of a better word, in that they can by apparant design define almost limitlessly how they relate to each particular group.
On the other hand, you might be nearer to my style even than that, in that you might be running an adventure path and a sandbox nearly simultaneously and switching back and forth as necessary. Two sessions ago I ran a nearly complete 'sandbox' that I think would have been very familiar to you. Everything I prepped had to do with player goals and events from their personal background. Last session I dragged the players back on the adventure path, which is the fight against a group who is at least for now the clear bad guys. Naturally, there was once again an example of my reoccuring theme of 'the chase', in this case, a scene inspired by the boat hopping chase in 'Mad God's Key'. As the players begin to get a feel for the situation, I suspect we'll default to more and more sandbox play with more and more player initiative (and I'll adapt less and less published material into my story line).
