Why do we have to exhale (or inhale) in order to speak?
As part of your question, you ask if other animals can create sound without continuous airflow. Many insects (e.g. cicadas and moths) do exactly this by using tymbals. A tymbal/timbal is an external membrane organ that is controlled by muscles or wing movements, that cause the membrane to flip back and forth, creating clicks or other sounds. So in many ways it produces sound in a similar way as loudspeakers, by creating sounds using a vibrating membrane. 

One example can be found in this clip, which nicely shows how the organ functions:
Moth Making Sonar Jamming Sounds (youtube)

On this page (in Swedish) you can also listen to slowed-down butterfly sounds from four species.