My interpretation of the final chess move scene was to show that Lena is a very calculating person who has her own goals that she is pursuing.
I think the chess piece was meant to show that she was reflecting on the moment from her childhood when her mother said to her "You may be a Luthor after all." The problem with TV today becoming so driven by plot twists is that viewers (and sometimes writers' rooms, unfortunately) often forget that what should matter most is what a scene means to the characters emotionally, not how it foreshadows or reveals some plot development. Lena has just learned that she's Lex's biological half-sister rather than his adopted sister. That's something she has to process emotionally, an upheaval of her whole sense of identity.
I agree that showing her beating Lex at chess definitely reveals that she must have a remarkable intellect. And yes, it could be meant to hint that she has some larger game in mind (although there's certainly nothing intrinsically sinister about being a chess player). But the emotional meaning of the scene is important too.
If anything the fact that it was so ominous makes me think it was just a misdirect, and she's actually plotting something good. They have spent so much time setting her up as a friend for Kara and ally of Supergirl I'd really hate to see them undo all of that just to create an unnecessary parallel with her brother and Kara's cousin.
Yeah. Like I said, this is at least twice now that they've made us suspect she was bad and had her turn out to be good -- I'd like to see that treated as a settled question and move on, instead of just repeating the same bit. I don't believe she's secretly evil; I think she is good, but the revelation of her biological Luthor-ness may potentially lead her to doubt herself and lose her way, in which case Kara/Supergirl will have to save her from herself.
I am glad that they put a fairly quick end to the Eve/Mon-El thing
I dunno, I liked it because we got to see more of Eve. She's really cute.
I do wonder why they even named her Eve Teschmacher, though. She's nothing like Valerie Perrine's character from the movie, and she has no connection to the Luthors (unless it turns out she's secretly been a mole this whole time). It seems like maybe they just put her in as a one-joke character so Cat Grant could yell "Miss Teschmacher!" as a Gene Hackman homage, but they liked the actress so they kept her around.
I was a little surprised that the rest of Kara and Alex's friends didn't know about Alex and Maggie.
I did like the bit establishing that Daxamites had no concept of homophobia, that Mon-El didn't think the gender of romantic partners mattered in the least. I wonder, does that mean he's bi?