Killer Croc and Man-Bat can die, while it's not required that you kill them it's difficult to avoid killing the crocodiles in the sewer level since they are enemies. You can also kill rats and maybe a few other animals that roam levels but no cats, dogs, horses, or dragons.
It’s revealed immediately after that he’s just pulling off a sick gag but Joker puts a gun to his head and pulls the trigger during the Arkham Asylum cutscene after the Joker campaign. The gun only fires a bang flag but it’s a shocking scene for a family game so I put it as a yes.
Only if you sympathize with the villains for whatever reason, it's possible that people died to Joker and Penguin's armies but this is never even handwaved.
There is a shocking reference to it in one of the purchasable "hints" (which eventually get to the game's lore) but the gallows mentioned were destroyed before the events of the game.
Oddly violent scene of Robin getting beheaded by a manhole cover Killer Croc throws at him before Batman reattached his head. The head is still alive but the neck hole is red and doesn’t seem hollow, unlike real LEGO heads.
Typical Batman trope of ableist stereotypes as villains. Poison Ivy and Catwoman, who are bisexual in the comics, are only able to kiss men in this game.
There is a part where you have to drop explosive stuffed animals on objects to destroy them. This is to say nothing of all the Minifigures and LEGO objects that get destroyed but it's debatable whether they count since they aren't toys in-universe and therefore don't belong to a kid.
The Joker mockingly pretends to cry over Commissioner Gordon's plight and could be imitating him, I'm not sure if this counts. Batman and Robin don't care when the Penguin cries but don't ridicule him over it.
Batman is a father in some universes and Commissioner Gordon is confirmed as one since Batgirl is in the game, anyone can die in this game but they all respawn. You can make Ra's Al Ghul in the character creator but his whole schtick is getting resurrected even ignoring the game's respawning mechanic.
There’s a gameplay mechanic where you can optionally save people getting beaten up by bullies in most levels. It’s not always possible to save them the first time you play the level.
I doubt anyone considers fear toxin and laughing gas drugs, not in that sense. Bane uses venom, as always, but it's not a real drug and you can only see the tubes by looking at the back of his model.