I gave this some thought and came to the conclusion that a continuous wave hitting a fractal would not have a defined reflection, since the fractal does not have a traditional derivative it would also not have a tangent at a point.
No, below a certain resolution (depending on the wavelength) the waves just see the average shape (which is smooth has a derivative). Think about this in real live imperfections in the atomic structure of a mirror don't influence the reflected light rays, or the shape of waterwaves reflected of a concrete wall is not influenced by the roughness of that concrete.
you are right, I've recently done some Points Billiard in Koch Fractals video which tries to do classical billiard in Koch Fractal which is exactly what you write about averaging etc. (I had to average the normals around hitting point as the normal vector was not defined here), see the video here: https://youtu.be/v9KS03hiSU8