The defining property of a conductor is that charge is free to move within it. Hence, if there existed an electric field within the conducting medium, charge would move until the field became zero. It follows that $\vec{E} = 0$ inside of a conductor.

Gauss's law therefore implies: $$ \rho=\epsilon_0\nabla\cdot\vec{E}=0, $$

since $\vec{E} = 0$ within the bulk of the conductor, all of the excess charge must reside on the surface.

The defining property of a conductor is that charge is free to move within it. Hence, if there existed an electric field within the conducting medium, charge would move until the field became zero. It follows that $\vec{E} = 0$ inside of a conductor.

Gauss's law therefore implies: $$ \rho=\epsilon_0\nabla\cdot\vec{E}=0, $$

since $\vec{E} = 0$ within the bulk of the conductor, all of the excess charge must reside on the surface.

To address your two questions specifically;

• In a metal, the electrons flow freely around like a fluid. They are not associated with any particular nucleus.
• The charges will do whatever they need to, in order to make the field zero inside. This defines how the charge acts on the surface. Your assumption that you know the charge distribution and from that you can determine the field is backwards.