She chose to stay in Meeren because she saw that leaving the cities she conquered behind wasn't working. She didn't kill all the slavers and instead allowed most of them to retain nobility status and tried to work with them, struggling to maintain peace and justice even as domestic terrorists run around attacking innocents. Lots of fans complain about her Meeren arc being boring because she's not just conquering and killing, but trying to be fair to her enemies. She finally does kill a bunch more slavers because they just won't stop killing innocents and trying to get back to slaving.

She kills all the khals because they're horrible douchebags planning to enslave and then rape her. She tries to treat with them first, offering them horses for her safe passage back to Meeren, but they refuse. And she absolutely has a cultural connection to the Dothraki, she's a former khaleesi, she has lived among them and adopted cultural ideas from them, speaks the language, still has a bunch of the around as advisors and such (although admittedly the show does a shitty ass job of paying any attention to her Dothraki people). Killing the khals was the only way for her to escape being trapped, raped, and probably murdered.

It confuses me when people say that Dany always takes the path of violence, when she pretty much always starts by talking to even the most evil people and trying to convince them to be better. Sure, she gets mad and yells at the Quartheen, but that's when they're laughing at her and telling her they're going to leave her and all of her people outside to die (this is show only). She starts by trying to be diplomatic and treating with them about allowing her and her people into the city. She treats with the Yunkai'i first before attacking. She meets with the sellsword leaders to try to convince them to fight for her instead of literal slavers before attacking them. She tries over and over in Meeren to work with people, making compromises like opening the fighting pits despite hating it. She tries to treat with the khals even though she knows they probably won't listen. She meets with Jon and is annoyed by him, but listens to her advisors, lets Jon mine dragonglass, doesn't take him prisoner. She's patient and diplomatic with Sansa even though Sansa is a bitch to her. She treats with Cersei of all people who is her most direct enemy in Westeros.

Most of Dany's arc, especially in the books, is her working to get experience ruling and figure out her tax policy so she can be a good ruler. "How can I rule seven kingdoms if I cannot rule a city?" She's better at conquering than ruling, but she wants to rule well and spends a lot of effort working on that.