How does the glGenerateMipmap function create the mipmaps? Does it do some kind of interpolation, does it take the average of 4 pixels (this is what assume), or does it skip every second pixel in the next mipmap level?
And is there a way to influence its underlying algorithm? Or choose the way mipmaps are created?
The OpenGL Spec, Section 8.14.4 states about mipmap generation:
No particular filter algorithm is required, though a box filter is recommended as the default filter
It is not possible to modify the algorithm used by glGenerateMipmap, but you can create mipmap levels on your own with any algorithm you like and then upload them to the texture using glTexImage2D(...) with the appropriate level. Level 0 is the full resolution, Level 1 is the one with 1/2 of the original size and so on.