Kate pretty casually confirms Wonder Woman exists in the Arrowverse, after many hints in the other shows. When and where she operates/operated is unclear. Themyscira, Wonder Woman's birthplace, was confirmed to have existed in DC's Legends of Tomorrow (2016).
The note Beth leaves Kate, "You have our father's eyes", is the line she used to reveal her true identity to Kate who was acting as Batwoman in the comics.
Vesper Fairchild talks about the incident on the bridge and questions exactly where Batman was during the incident, and if he was at "Robin's high school graduation". While it was probably safe to assume that Earth-1's Batman had had a Robin at some point before his absence, this line is the first time that he's actually been referenced within the Arrowverse. This name-drop comes at an interesting moment for the Arrowverse as a whole, particularly in relation to the upcoming "Crisis on Infinite Earths" crossover. Burt Ward, who portrayed Robin the 1960s Batman TV series (Batman (1966)), will be in the Crisis on Infinite Earths crossover event.
Jacob orders Alice to be sent to Arkham Asylum, first featured in the Elseworlds crossover and of course an important part of the Bat-lore.
The underwater scene was a 10-hour camera day on a Saturday.