If a soundproof door is almost shut but still leaves a small opening, what wave phenomenon is responsible for someone being able to hear sound from the room while standing by the wall to the side of the opening?

The phenomenon which is responsible for this effect is called diffraction.

Diffraction is the ability of a wave to propagate when it meets an obstacle or a slit. When the wave encounters the obstacle or the slit, it 'bends' around it and it continues propagate beyond it. A classical example of this phenomenon is when a sound wave propagates through a wall where there is a small aperture (as in the example of this problem)

it is called kinetic energy