All of this is related to "skin effect". Basically, for high frequency, the magnetic shoves the electric field to the edges and surfaces of the conductor. Do a search on "skin effect", and you will find lots of Vector math expressed with Maxwell's equations.

The second part of your question, about the charge density on the metal rod, does not have a simple answer. The charge density is highest where the electric field is highest and the electric field is highest at the pointy ends of the rod. I don’t think you can get a simple equation for the answer. It is probably an integral with no closed form solution and would normally be solved nowadays by direct simulation.

Static charge distribution has nothing whatsoever to do with skin effect (a dynamic phenomenon.)