Does the focal length change? Or does it remain constant and only the brightness of the image is reduced?
you will end up with 2 lenses at different locations and some light will go to 1 lens, some to the other and some to both resulting in 3 images in 3 different planes - so yes the focal length will change for the 2 new images.
Each lens piece will have a lower aperture than the full lens and therefore a lower image quality.
In microscopy we often hide part of the incident light (see oblique illumination)- which amounts to the same as using a section of the lens- for increasing contrast at the expense of definition.