A Lot of people might be initially confused by airi's odd attraction to tenchi and even to Seina. after all, she is a married woman, and tenchi's grandmother and clearly a lot has been lost in translation.
These are my personal observations on the subject and honestly nothing has ever been said point blank about this but this is what if figure makes the most sense.
on the subject of tenchi: Tenchi looks like a young yosho. Aeka noticed. it in episode 3 it is safe to assume that airi noticed it too, Tenchi also looks like his mother/airi's daughter Kiyone when she was young. Airi had sex with yosho once and got pregnant in her very early 20's it's entirely possible yosho was her first. and then didn't talk to her for about 80 years or so...
also the way tenchi talks in Japanese when she swoons over him is well... accented very handsomely.
but more than that just like yosho, who made the mistake of calling her Magma-dono the first time he addressed her tenchi messed up and called her grandma.
So when you meet a guy who looks ridiculously like your first love and makes a mistake just like he did... it makes sense she is acting a bit weird.
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With seina honestly it's a bit more of a wild guess.
you see she hit on him before he captured widow but airi just seems to be a flirt in general. I mean seina is a freakin bajillionaire so there's that...
After book two though she gets very very very sexually aggressive towards seina... why?
the obvious reason should be obvious. Seina captured widow. widow killed her father and destroyed the nation of airai she had a massive vendetta against her and seina captured her and got her sentenced to death. after that happened she practically throws herself at seina.
not to mention she may be married to yosho but it's a secret common law marriage and he is light years away. airi is overworked and under loved.
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So why does mikami go after him? well in the books airi and seto have a very civil but intense rivalry as well as having a pretty close friendship. that's one reason, but another is stated to be that when widow gave her report of how seina "captured" her she tells mikami about how seina's purity is like poison to people like her. and after then mikami tells Airi that after talking to widow she is thinking of putting herself back on the dating market.
That's the best i can figure out for now as i'm not actually far enough into translation to come to any scene where she tells seina why she likes him.