Yes, but then he re-wrote reality again and Kara still wasn't affected
Yes, she was; she was teleported into a Pipeline cell in transformed clothing, and given a new identity as a wanted criminal. She, Barry, and Oliver all remembered their real identities, so it wasn't about what Earth they came from; presumably Deegan wanted them to remember. But they all had their circumstances and their identities in the new reality transformed along with everything else.
in fact didn't Deegan make a big deal about that fact when he confronted her in the Pipeline.
Yes, about the fact that the Book of Destiny didn't contain information about her and Superman. That had nothing to do with immunity. It probably didn't contain information about Sherloque either, but he was changed along with everyone else because he was there at the time.
Yes, but I would assume she would have also seen their memories?
Yes, as I already said, Cecile's mind-reading would only confirm that they fully believed they were each other. By itself, that wouldn't prove that reality had been altered, only that their minds had been.
I think the truth is that Oliver was just in denial about Batman's existence.
It's not just Oliver, though. Early seasons of
Arrow treated Oliver's vigilantism as a novel practice, and the crossover episodes have always treated the Green Arrow as the senior member of the Earth-1 hero community, the pioneer the others look up to. That only works if Batman's existence is not common knowledge. Putting him on Earth-1 at all is an imperfect retcon, and establishing him as an urban myth whose reality is not widely accepted is the only way to patch it over at all. (I still think it would've been better to put
Batwoman on Earth-38, but that would've required a massively different approach to
Elseworlds.)
As for the Penguin, wasn't in Arkham in a recent Batman story or animated adaption? Like you I've never considered the Penguin a lunatic, infact I see him as a bridging character between the likes of Falcone & Morone on the one hand and the Riddler etc on the other.
I don't recall anything putting Penguin in Arkham, but it's possible. Modern Batman stories have become overly enamored with Arkham to the point of sometimes putting characters there who shouldn't be.
Batman: TAS was ambivalent about whether Killer Croc belonged there ("Trial") or not ("Sideshow").
Something that's really bugging me is the Earth-1 people knowing of and using Kryptonite. When Ollie shot that Kryptonite arrow at Nazi-Kara, it seemed to be a one-off and as such people seemed to let it go with minimal questioning. But now its been used multiple times in the same story and Kara's 'Unless someone has Kryptonite lying around' was 'Chekov's Gun' being cocked if I ever heard one, they're going to have to explain it.
I don't recall any kryptonite being used here except by the Green Arrow, so that merely perpetuates the existing mystery of how he got it rather than adding to it. And Kara's kryptonite crack clearly wasn't a Chekhov's Gun, because it wasn't followed up on. If anything, it was the opposite; Kara was pointing out that if Kate Kane didn't like her patrolling the city, there was nothing she could actually do about it.
I was also a little frustrated about the fate/backgrounds of Earth-1 Kara Danvers & Jimmy Olsen.
If you mean we never found out what their real lives were like, yeah, that's a lingering question. As I remarked before, it's kind of sad to think of them never meeting Kara and Clark. Alex's life was probably profoundly different with no Kara, no J'onn (presumably), and no DEO (as we know for a fact from "Invasion!"). Maybe she became a doctor instead of an agent. Jimmy might not have turned out too differently, though; if Metropolis, Lois, Perry, and the
Daily Planet exist, he probably would've still had a good career as a photojournalist (although Lois might've gotten herself killed ages ago without Superman to save her from her intrepid overreaches).
