A square
Yes it can.
Infinitely many.
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
No, a cube cannot have an octagonal cross-section.
A square.
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
equilateral, isosceles, and scalene.
You would cut off a corner.