From the way she's been treated in the last couple of episodes, I'm not liking how they're turning Caitlin into a joke.
Also, on the question of whether or not Earth-1 has Batman: there's the Arrow episode in which Oliver gave Felicity her "Overwatch" codename, and said, "I was going to go with 'Oracle', but it's taken" - indicating that Batgirl exists and is paralyzed, so the Joker exists, so Batman exists.