It made sense that there would be some kind of clash between them, since they're the two most opposite characters, in terms of both personality and placement on the grounded/fanciful spectrum. Oliver's had to cope with a lot of weirdness in his life, but lately, in the wake of last season's magical catastrophes, he's trying to get "back to basics" and ground himself again (as is his show), and now he's confronted with aliens and parallel dimensions, and Kara embodies both at once. So she was a bit much for him to take. Conversely, Kara's just been dealing with anti-alien bigotry on her own world -- in a pretty horrific way with the Medusa-virus mass murder -- so that explains why Oliver's reaction made her so angry (though I wish we'd gotten a scene making that explicit).
Speaking of things left out on Kara's end, it was weird that she said the hero community on her world was "just me and my cousin." She should've mentioned J'onn, at least, though I can understand her still being unsure about the Guardian.
I think it's incredibly goofy-looking. And I'm usually okay with superhero costume elements that others find goofy.
It's another thing that implicitly makes sense but would've been nice to see spelled out more. Thea's retired from crimefighting because it was too traumatic for her. She was drawn back in by the temptation of facing aliens, but then she had an emotionally wrenching experience with the illusion of her parents being alive and having to give that up. I can see her deciding that she just couldn't take any more -- and that her skills were kind of redundant to the effort anyway.
Probably best not to think that through too carefully. (Though there was that creepy Silver Age comic where Superman expressed regret that Kryptonian custom wouldn't let first cousins marry, because Supergirl would be his perfect mate otherwise.)
Anyway, I'm surprised at how cavalier Supergirl was about giving away her civilian identity on Earth-1. That's a lot more people who know her secret now, and given how porous the multiverse is becoming, there's the possibility that information could get over to her world somehow. Although it's also pretty much an open secret at the DEO.
I was hoping for more Sara/Kara interaction. I'd read that Sara would be attracted to Kara and maybe flirt with her, but all we got was one line of Sara saying it was kinda hot what a badass Supergirl was. More seriously, given that Kara is adjusting to her sister coming out, it would've been interesting to see a nice quiet conversation about the subject between them, maybe learn more about Sara's own coming out.
I can see why they didn't go that route. It could've confused viewers -- Flash audiences are familiar with doppelgangers by now, but the viewers of the other three shows might've been thrown off -- and Marsdin's "secret identity" would create too many complications.
Besides, I like the idea that there are now two female presidents in the two featured worlds of the multiverse -- and Earth-1's POTUS is a black woman, which is even more of a potent statement now.
Maybe Kara will send them the blueprints for her DEO headquarters, so that the Earth-1 DEO can use the same set.
"In the future," heck -- I'm planning to marathon the whole thing today or tomorrow.
Sounds like you missed something. We saw in the Western episode how Ray got a new supply of dwarf-star ore, as well as how the Waverider's fabricator could quickly create any desired costume or equipment (a fact also mentioned in the Arrow portion of "Invasion!" to explain how Team Arrow got their gear back after escaping the Dominators).
