You are somewhat mistaken. You are thinking of the skin effect. In DC power, current flow is proportional to the cross section. As the frequency increases, the effective resistance in the interior increases relative to the surface. At high enough frequencies, conductors effectively conduct only along the surface.

See https://en.wikipedia.org/wiki/Skin_effect and https://en.wikipedia.org/wiki/Litz_wire .

Thank you, looks like I was mistaken :D

No, generally a thicker cable is a better conductor because its resistance is lower. If it were merely a matter of surface conductivity this wouldn't necessarily be true. However, internal conductivity and surface conductivity can be affected by magnetic fields:

http://faculty.wwu.edu/vawter/physicsnet/Topics/Gauss/IdealC...

There is also a tendency for currents to creep towards the surface of the conductor at high frequencies, see [1], although this is ultimately also caused by internally generated electromagnetic fields.

[1] https://en.wikipedia.org/wiki/Skin_effect

For static charges, all of the charge on a conductor will be at its surface:

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gausur.h...

That doesn't apply to moving charge though (ie. an electric current)

You may be thinking of the skin effect associated with alternating current systems. https://en.m.wikipedia.org/wiki/Skin_effect