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
The cross-section of a cuboid is unified in the shape of a square or a rectangle.
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
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.
Along any edge
Cube, Cuboid. Any prism with a quadrilateral cross section
You would cut off a corner.