I believe if you were to travel around Teyvat, you'll find how each region functions is very different from even their neighboring areas. How the Archons decide to lead, if they do, is very different. [From the way Mondstadt functions without their Archon present to Inazuma's Shogun very much in charge of her nation.] I think we all have some criticisms about how things are ran, but she is well loved by her people.

[Furina truly was loved by Fontaine, he thinks.]

Both. I spend much of my time dealing with things that need my attention, but you can also say that I feel more at home there than in the overworld.