As with the series this animated film is based on, much stylized and humorous fisticuffs (complete with "Pow!" "Biff!" "Zap!" overlays) are on display during the setpiece fight scenes.
Batman attempts to kill Robin and Catwoman by placing them in the Batcave's nuclear silo (a reference to the death of a character in the very first two-parter in the original show's run, one of the only times where any character was known to have definitively died). They escape from the predicament.
Catwoman is shot out an airlock of a space station, but is shown to have had no ill effect whatsoever to the exposure to hard vacuum.
Batman and Robin are threatened with death in a elaborate deathtrap of the kind routinely used in the original series (this time they're tied to a gigantic TV dinner that is slowly being moved towards an gigantic oven).
Catwoman falls to her apparent death by apparent suicide rather than be captured by Batman again.