All about Eve Teschmacher

Being much more straightforward about it, I'd say it's possible she or Lex knows Kara is Supergirl, simply because Lillian Luthor knows. We know she didn't tell Lena, because she believes that secret will, one day, backfire on Kara. But it makes perfect sense to me that she would share this information with Lex.

The writers didn't know until this season that they'd even be allowed to USE Lex. Eve, most likely, was originally just a comical easter egg referencing the Superman movie's "Miss Teschmacher". But, after learning that they would be able to use Lex, turning Eve into a villainous pawn of Lex makes a lot of sense. She's a recurring actress they've had on for a couple years, and it gives her a bit of opportunity to actually make something more of that character.

And as you're pointing out, it makes us go back and look for possible nefarious intent in her previous actions since season 2. It's certainly possible to fit her previous actions into her new nefarious role. But it was kinda clear that they were gonna do something more with her, this year. What she's been doing with Lena, and with what we've been learning about how smart she really is, shows just how overqualified she was to be an assistant at CatCo, I'm actually shocked I didn't see her heel turn coming sooner, because it was pretty clear foreshadowing. It should have been incredibly suspicious for this meek assistant at a media company studied nuclear physics and had engineering knowledge, lol!

But again, I can admit I didn't put it all together until someone here mentioned they thought Eve shot James. Then it made perfect sense.

But yes, even if Lillian didn't tell Lex who Supergirl is, I don't think it's out of the realm of possibility that Eve could have figured it out.

I agree Kdogg when Eve Teschmacher was cast in the beginning of season two it was an Easter egg, TPTB had no idea she would be used this way. At the end season three they decided to do something with the character but I still don’t think TPTB anticipated this exact turn of events. It’s to Andrea Brooks’s credit that from the beginning she’s played Eve as someone with more going on under the surface which makes this turn of events believable. My gut says they got permission to use Lex after season four started filming and fit him in the story. I think the Lex storyline is taking the place of the originality planned Winn storyline. Gavagai what you said is a great in show explanation.

Wasn't it Kara who recommended Eve as her replacement to Cat though?

Eve didn't start out there and wasn't even guaranteed to be Cat's assistant since no one knew that Kara was going to get promoted in order to open up the spot or that Kara would recommend that Eve take her place. Seems like a lot of "if, and's and but's" that would need to line up for it to be an intentional placement.

I'm not saying it was a sure thing, but I can see it as something Lex would consider worth trying. And if it didn't work they would find another way for her to cross paths with James.
Kara was probably given a pile of applications to sift through. I wouldn't put it past Lex (or Lillian for that matter) to have the connections to make sure Eve's was near the top. He might have even dug up info on Cat's last assistant, and Cat's preferences in general so he could coach Eve how to ace her interview. That could explain why Eve adopted a meek persona similar to Kara's she even dressed similarly.
As far as TPTB are concerned, I'm not all that fussed about what TPTB intended. I really only care about what they actually do. But yes it's unlikely they would have been able to keep it secret for that long, so it probably wasn't all planned out since season 2.

Wasn't it Kara who recommended Eve as her replacement to Cat though?

Eve didn't start out there and wasn't even guaranteed to be Cat's assistant since no one knew that Kara was going to get promoted in order to open up the spot or that Kara would recommend that Eve take her place. Seems like a lot of "if, and's and but's" that would need to line up for it to be an intentional placement.