Question

Does electric current (both dc and ac) flow through the surface of a conductor or through the space in the conductor??

Answer 1

It depends on the frequency. DC electricity travels through the bulk cross section of the wire.
A changing electrical current (AC) experiences the skin effect where the electricity flows more easily in the surface layers. The higher the frequency the thinner the surface layer that is usable in a wire. At microwave frequencies the depth of the metal that the current flows in is about the same as a wavelength of visible light