Yes, but not a regular one.
Along any edge
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).
the type of shape is a square A Square, An Equalateral Triangle, A Cube, A Hexagon, A Pentagon, A DoDecaHedron, An Octogon, A Decagon. A Square, An Equalateral Triangle, A Cube, A Hexagon, A Pentagon, A DoDecaHedron, An Octogon, A Decagon. A Square, An Equalateral Triangle, A Cube, A Hexagon, A Pentagon, A DoDecaHedron, An Octogon, A Decagon.
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.
No, a cube cannot have an octagonal cross-section.
A square
A square.
Infinitely many.
Yes it can.
Rectangle
The answer depends on the inclination of the intersecting plane. The cross section can be a point, a triangle, a rectangle, a pentagon or a hexagon.
It will be the same shape as a square
It is a prism with a pentagonal cross-section
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