* Does every convex polyhedron has a [combinatorially isomorphic](http://en.wikipedia.org/wiki/Convex_polytope#The_face_lattice) counterpart whose all faces have rational areas?
* Does every convex polyhedron has a combinatorially isomorphic counterpart whose all edges have rational lengths?
* Does every convex polyhedron has a combinatorially isomorphic counterpart whose all vertices have rational $x,y,z$ coordinates?

Can multiple conditions above be combined?

Update: all polyhedra in question are in $\mathbb{R}^3$.