Do all polyhedrons have more edges than vertexes?

Answer 1

Yes.

to add to that

a vertex must be connected to at least 3 edges to be 3d, an edge is always connected to 2 vertexes, so the closest the two can ever be is

vetexes x 3 = edges x 2,

but when working with any platonic solid you can follow this:

vertexes x (faces / vertexes) x [edges on one side] = edges x 2

or

vertexes x [faces meeting at one vertex] = edges x 2

when working with any other polyhedron

[vertexes with x amount of faces] x (x) + [vertexes with y amount of faces] x (y) ...{and so on} = edges x 2