Yes
No. Not even all hexahedrons are cubes.
Yes. It is one of the five regular polyhedra known from ancient Greek times or earlier. See http://www.math.rutgers.edu/~erowland/polyhedra.html .
(a3-b3) = (a-b)(a2+ab+b2)
The sides of a rectangle are not all equal so a rectangle is not a regular polygon. And, as a consequence, any 3-dimensional shape made from rectangles cannot be a regular polyhedron.