Suppose a hypothetical single, non-spinning star. Does such a star produce GW?
No more than a nonmoving boat floating in a perfectly still ocean produces water waves. Gravitational waves are produced by things actively disturbing the spacetime around them.
More technically, there needs to be a time-varying system with nonzero quadrupole or higher moment. A perfectly spherical object expanding and contracting does not produce gravitational waves, nor do two point masses moving along a line toward and away from each other. However, two masses orbiting each other have a nonzero quadrupole moment and so produce waves.
I assume GW do not come from inside of a black hole
Indeed most of the signal we see clearly does not. The initial "inspiral" portion of the waveform (where the amplitude of the wave is slowly building up) is generated when there are two distinct black holes orbiting one another. It is the combined system, not either black hole individually, that generates the waves.
After merger, we see the "ringdown" phase (where the waveform rapidly diminishes back toward the noise floor). This pattern is produced when the event horizons have merged, so there's only one black hole. Still, the waves don't have to "escape" from inside the black hole. At this point, the black hole has not settled down into a stationary state. The horizon itself, and the surrounding spacetime, are in some sense still vibrating.