Conductivity in a metal results from the metal atoms having (1) high electronegativity (2) high ionization energy (3) highly mobile protons in the nucleus (4) highly mobile electrons in the valence shell

The answer is (4). You may recall hearing about the "sea of electrons" model of metals. Metals represent the ultimate case of delocalized (shared) valence electrons, and these delocalized valence electrons are what freely move around, conducting current through the body of the metal.