"The Japanese girl does not lie down."
Translation:A japán lány nem fekszik le.
In this case it is nem fekszik le because the negative nem needs to be immediately adjacent to the verb. If you're negating a verb, nothing can stand between the nem and the verb, not even one of the preverbal particles like le-, ki- or be-.
More generally, the answer is "the particle gets detached and moved after the verb if there's something more important that needs to be right next to the verb." But I think you will only develop your sense of when that happens by seeing lots of examples.
Lots of exceptions, but a good rule of thumb to start with, is "negative statements and questions" use a separate preverb, and all other sentences (mostly) do not.
I'm not a native speaker ... I'd appreciate it if a native could confirm that this is roughly accurate?