Atomica manipulating Superman into killing Dr Light in Trinity War - Ultraman's weakness is sunlight, so naturally you'd eliminate the one member of the three Justice Leagues who's a literal walking solar battery before Ultraman's arrival.
Platinum of the Metal Men's malfunction in her first appearance in Justice League #18 becomes much more understandable given that Atomica was the one who most likely sabotaged her Responsometer.
Although he has yet to meet Superman, the New 52 incarnation of Bizarro introduced in Forever Evil is an excellent Foil to the Man of Steel instead of simply being an Evil Counterpart. While Superman is a mature adult with a good moral compass, Bizarro is a Man Child who has not yet developed a sense of right and wrong. Bizarro is not evil, he is just following the orders of his creator, Lex Luthor because he sees him as a father figure. This is in contrast to the pre-New 52 version of Bizarro, who was portrayed as a consciously evil creature with a bad understanding of grammar.
Grid's inability to find anything strange about Alexander Luthor/Mazahs: since he's powered by magic, Grid's technology simply couldn't find anything abnormal about him, because technology can't detect magic, as far as we know.
Similarly, his inability to find anything about Superwoman is due to her falsified reasons for being there. Why would she give up that her goal is to set Alexander free and be with him and their baby with an AI on their side?
Lex Luthor's goal in life has always been to destroy Superman one way or another. In the final issue, Alexander Luthor kills the Superman clone Bizarro and Lex Luthor spares the life of Superman's Earth-3 counterpart Ultraman while belittling his overreliance on his powers. Luthor's Heel-Face Turn wasn't just for the sake of it. His confrontation with Alexander Luthor was him realising what a monster he had been over the years and he wanted to break away from it.
The reason Black Adam and Sinestro both reject the Cut Lex Luthor a Check option could be due to their villainous tendencies... Or it could be that as the former isn't a US national and the latter is an alien, a pardon by the US president means absolutely nothing to them.
In a flashback in Justice League #26, an offscreen Alexander Luthor states that he hopes to save the world from the Crime Syndicate and to stop the "monster" they lured there. When he is finally released he has failed in the latter, but he is determined to not fail in the former and his only option is to gain as much power as he can to stop the Syndicate and avenge Earth 3, by any means necessary.