A square
Yes it can.
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.
A square
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
Oh, dude, calculating the cross-sectional area of a cube is like a walk in the park. You just take the length of one side of the cube and square it. So if the side of the cube is 4 units long, the cross-sectional area would be 16 square units. Easy peasy, right?
You would cut off a corner.