A square
Infinitely many.
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.
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 cannot unless you know the angle of the plane which defines the cross section.