In the Lego Cinematic Universe, people don't seem to die if their bodies get cut in half.
Is there any way for them to die?
Yes, seemingly, by jumping off a cliff.
In The Lego Movie 2: The Second Part, two characters scream "I don't want to die", and then fall off a cliff. This implies two things:
They can die (otherwise, there's nothing to get scared about),
Falling off a cliff kills them (though it's possible that this part is merely played up for comedic intent, because other characters survive longer falls. Yet it is heavily implied they die.
Absolutely; one character,
dies at the end of The Lego Movie.
Characters surviving extreme injuries comes down to (TV Tropes Warning) The Rule Of Funny:
Any violation of continuity, logic, physics, or common sense is permissible if the result gets enough of a laugh.
In The Lego Movie 2: The Second Part, we see the timeframe altered and
fade from existence Back to the Future-style. One can argue whether or not this character really dies, as in reality they are
An embittered, vengeful version of Emmet from the future,
so it's debatable whether they truly have their own independent "soul" that can be killed. One could easily argue either way - that yes, traveling to another time in which you also exist "splits" your soul (e.g. so that "You" and "You Prime" become independent entities that each can live or die), or no, that there's just some strange quantum entanglement or whatever going on. In the end it's left open.