It depends on the angle of the plane of the cross section.
If it is parallel to the cube's face (or equivalently, two adjacent edges) the cross section will be a square congruent to the face. If the plane is parallel to just one edge (and so angled to a face), the cross section will be a rectangle which will have a constant width. Its length will increase, remain at a maximum level and then decrease. If neither, it will be a hexagon-triangle-hexagon-triangle-hexagon (triangles when passing through a vertex).
Yes, but not a regular one.
Along any edge
length = volume/cross-section
The vertical cross section of a right vertical cone is a triangle if that cross section is taken from the vertex. Any other vertical cross section will reveal a hyperbola (with endpoints on the base of the cone). A link can be found below.
You cannot have a 2d cylinder. The 2d cross section will depend on the plane of the cross section.
No, a cube cannot have an octagonal cross-section.
A square.
A square
The cross-section of a cube can vary depending on how it is cut. If sliced parallel to one of its faces, the cross-section will be a square. If cut diagonally, the cross-section can be a rectangle or a more complex polygon, depending on the angle and position of the cut.
Yes it can.
Infinitely many.
Rectangle
Yes, but not a regular one.
It will be the same shape as a square
The following are some shapes having a square cross section: a cube, a cuboid, a square pyramid.
Cube, Cuboid. Any prism with a quadrilateral cross section
Along any edge