Something I worked out is that a character is well created (or, for a fanfic writer, well understood), if you, the writer, can drop the character into a situation, watch what they do, and think, 'yes well of course, being X, what else could he do?' If doing that leads to a character reacting in a totally unexpected way, either your understanding of your character deepens, or shows an unexpected facet; otherwise, you know you have it wrong, and will have to take that character back to basics, and build or discover them again.
This, definitely. For me, the moment I know I have a fully-fledged character on my hands is when I want them to do something only to realize that this is a thing they'd
never say or do.
As for the question of whether a character can exist independent on the setting, I'd say that the part that stays the same regardless of what situation the character comes from is very, very small - if Tuuri had grown up going on expeditions to Silent World every Tuesday, would she still be so excited for the outside world? While she may still long for being "out", her attitude and reasons behind her wanderlust would be completely different from canon Tuuri's. Would Onni, with his apparently quite impressive powers, still be so scared of the outside world if he was Immune? Probably not, though he might still fret about unexpected problems arising.
For me, the reason AU characters end up looking so similar to their canon versions in many fan-fics is that, consciously or not, the writer tweaks the situation they're in to be similar - or equivalent - to that of the canon versions. A fun thought excercise: if Emil and Lalli were swapped at birth, would they still become friends when they met in year 90?
So my question would be: Do you think a setting can exist independently from a character? Would you be able to use the same setting for two stories or as habitat for completely unrelated sets of characters?
I think it's called "expanded universe"...
But to answer the question, this depends on how much you've created of your world. I've recently became quite a fan of worldbuilding - I have an entire world map, with all countries and their capital cities marked, for a story which never moves out of a hundred kilometres' radius from the start point! By now, if you pointed at any random point on the map, I could give you a story set in it. It helps if you approach it sorta like building an RPG campaign (which is how the abovementioned story started). You set up a coherent setting, give it rules it abides by, figure out more important parts of its backstory and present-day events, and voila! It can accomodate pretty much every character.
BTW, have you read Brandon Sanderson's Cosmere books? That's the kind of a thing I'm talking about. With the amount of building that went into it, everyone could create a Cosmere world, magic system and characters and never have them interact with canon characters or planets - and it'd still be recognizably a Cosmere story.
Of course, on the other hand, there are characters so powerful that the setting would look completely different if you were to take them away, but that's probably not someone who'd makes for a good main character. See: the God-Emperor of Mankind in 40K. If you plucked the guy and replaced him with someone caring, pleasant and... let's call it "psychologically canny", the setting would be completely turned on its head. In a way, it's like a "noble vs grim" distinction I've seen once. A "grim" setting is the one where the world is completely immutable, regardless of what the characters do - to continue with my 40K example, no matter how many decisive victories the God-Emperor's side gets, the Imperium is still dying. The "noble" setting is the one where a few men can change the world completely - like in
Star Wars, where taking down the Emperor puts the galaxy-spanning Imperium on its knees and gives way for stuff like New Republic and First Order.
...oof, I think I've given you a small essay. tl;dr: characters change depending on the setting, but a well-built setting can carry different characters without changing, unless those characters aren't too powerful.