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First of all, with Nya being the older sibling and the "responsible" one, I think she'd be more mature in character. She still has her temper ofcourse and her impatience, but as women tend to mature faster than men, she'd also act more "parent-like" to younger Kai.

As Nya is the master of fire here and one of the OG ninja, her battle against misogyny starts much earlier than in Canon. The ninja themselves were a little bit influenced by society, so she'd be set on trying to prove herself as someone more than just "the girl".

Kai in the meantime wouldn't have any problem fitting into the team after becoming the master of water, society immediately accepting him as if he was always there.

THAT makes Nya even more devastated, as she had worked HARD to prove her place in the team and as a hero of ninjago, while her brother got accepted just like that, because of his GENDER. I think that'll be a big key point to her character. She wouldn't necessarily hate Kai ofcourse, pushing the anger towards the people.

Now about how Nya would feel with the green ninja thing:

As Nya wishes to prove herself, she sees the green ninja title as a chance to make everyone recognize her once and for all. Kai in the meantime knows he has zero chances of being the green ninja, so he gets over it quickly, rather focusing on how he could be useful while not having an element. (Yet.)

Later on when Kai gets told he needs to learn the element of water, a power inherited from his mother, he FLIPS THE HELL OUT.

Kai still has his fear of water, and now being told that he's supposed to be the MASTER of it? He'd rather be samurai X for the rest of his LIFE

Kai has a hard time to get over his fear, but thinking on how he could help the ninja finally after being "useless" before (in his own standard), he forces the fear to the side. He hasn't gotten over it completely until much later, but to save possessed lloyd he was willing to let it slide.

Kai doesn't remember his parents, as he was literally two years old when they disappeared. Nya herself doesnt know what to think of it over the years. Did they abandon them? Did something happen to them?

Too difficult as to what of think of it and what to tell kai over the years, she simply avoided the topic. Kai felt how it was uncomfortable for Nya to talk about, so he kept silent about it too.