The rank of Fundamental Matrix

This question is regarding two view geometry where a point lying in the image plane of the first frame/ position of the camera is mapped onto the image plane of the second frame/ position of the camera. This mapping is done with the help of Fundamental Matrix, $$F$$.

The rank of the Fundamental Matrix is 2 and I am trying to understand why. I have seen this question and answer. But that's not the perspective I want to discuss.

The book Multiple View Geometry in Computer Vision says:

Geometrically, F represents a mapping from the 2-dimensional projective plane $$P^2$$ of the first image to the pencil of epipolar lines through the epipole e . Thus, it represents a mapping from a 2-dimensional onto a 1-dimensional projective space, and hence must have rank 2.

1) My understanding was that the pencil is a family of planes passing through epipoles which rotates about baseline depending on the 3D point for which it is considered. What is a "pencil of epipolar lines"?

2) Depending on the point considered in a 2D plane, in the first image plane, the Fundamental matrix will map it into a line in the second image plane. But different 2D points can be mapped to different lines. So, the 2D projective plane is being mapped into a set of lines, not just one line. Why is that a mapping to 1D?

3) How does this change in dimension relate to the rank? Shouldn't rank be related to the output dimension?