A square
A cube cannot be a rectangular prism, as the cross section of this prism would be a rectangle, as opposed to a square. A square prism (if that is what it is called) could be a cube, but is not necessarily.
Infinitely many.
Yes it can.
It will be the same shape as a square
No, a cube cannot have an octagonal cross-section.
A square.
A square
A cube cannot be a rectangular prism, as the cross section of this prism would be a rectangle, as opposed to a square. A square prism (if that is what it is called) could be a cube, but is not necessarily.
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