Prosecutor Angus Stanton prosecuted Jack Napier AKA The Joker, which was also the real name of the Joker in Batman (1989). While The Joker's real name is usually an ambiguous fixture of DC Comics lore (something that the 2019 movie Joker (2019) acknowledges), several iterations of the character have used the name "Jack Napier" as some sort of alias. The name, which is a tribute to actors Jack Nicholson and Alan Napier, was originally created for Tim Burton's Batman movie and has since been used in Batman: The Animated Series (1992), 'Gotham Noir', and 'Batman: White Knight'.
Luke Fox researches the three kinds of punishment used - one of which was implemented in Gotham under "Mayor Cobblepot". This confirms that Oswald Cobblepot/The Penguin exists to some degree within the Arrowverse. Beginning in the New 52 era, Cobblepot was elected the Mayor of Gotham City, and used his power to benefit himself and his fellow super villains. This idea of Cobblepot being mayor has extended into 'Batman: Earth One', Gotham (2014), and Danny DeVito's portrayal of The Penguin in Batman Returns (1992).
The Executioner first appeared in Detective Comics #191 1953, where he made his only appearance. In that issue, The Executioner was Willy Hooker, a carnival worker who used that identity to murder wanted criminals and then, in turn, claim the reward for their capture. It also turned out that Hooker was responsible for breaking those same criminals out of jail in the first place, essentially setting himself up a lucrative, if brutal, little racket. He is ultimately discovered by Batman and Robin who put a stop to his activities.
Camrus Johnson explained in a interview with Entertainment Weekly that he was fully surprised about this twist in his character's story as compared to the comic books. "[I was] fully surprised because in the comic books, Luke is keeping his job and so many secrets from his dad; his relationship with Lucius is so important to the Luke Fox that I read in the comic," Johnson said. "So to start this show off with Lucius being dead is very interesting because there's a whole different side of the Luke Fox that I knew. It makes a brand-new Luke Fox because his icon, the dad he loved so much and wishes that he could protect, and the person that was his link to Bruce Wayne and Batman in the beginning, [is] not even there anymore. It was a total surprise to me, and I'm sure it's now a total surprise to the fans to be able to see why Luke is where he is and why Lucius isn't around."
Alice sticking her tongue through the false face's mouth was a "semi-joke" take, by Rachel Skarsten that made it into the final episode.