The show's Insistent Terminology always refers to Robin's real identity as Bruce's "youthful ward, Dick Grayson". It's an odd choice of words in a stiff formal phrase that seems out of place in the lunacy of the show, until it hits you that Robin's actor is Burt Ward. It's not just a phrase over-dramatized for comedy. It's an intentional pun on the actor's name...which has taken us over 50 years to get.
The Joker is a narcissist with a sick sense of humor, so it's fitting that his once-an-episode Villainous Breakdown moments are epic tantrums, like the one when he thought he and his henchwoman found the entrance to the Bat-Cave when all he found was a miniature insteadnote Might have been payback for the time he kidnapped the "Majarajah of Ninpah"
Crisis on Infinite Earths (2019) establishes that the show is canon on Earth-66, but it also provides a neat explanation for the conflictions caused by the comics, animated films and the Legends of the Superheroes specials. Much like how the main universe of that crossover was altered, it's likely that the animated films and the specials were canon pre-Crisis while the comics, which crossed over with Wonder Woman (1975) which in turn was established to be canon on Earth-76 by Crisis, are canon post-Crisis.
In "The Joker Goes to School/He Meets His Match, The Grisly Ghoul" Joker hooks up the Dynamic Duo to a pair of Electric Chairs, and if not for a handy power failure, we would've seen them fried.
By the same token, a lot of the villains' death traps, though whimsical and silly in concept, would have left a pretty horrific mess of our heroes' corpses if they had actually been successful.
In one episode, Catwoman's deathtrap-du-jour is designed to more or less lobotomize Batman, with the implication that she plans to keep him as a sex slave afterwards.
Even though this is a light-hearted Batman, he is still Batman and watched his parents get killed. The pilot episode has Bruce Wayne outright confirm that he's motivated by his parents being killed by criminals.