Electrostatics and electric field inside conductor

how can one have a negative surface on the cavity surface area and a positive surface on the outside area of the conductor

This can be achieved by moving positive charges from the cavity surface to the outer surface. Since positive charges are removed from the inner surface, an excess of negative charges will remain there; meanwhile, introducing new charges at the outer surface crates an excess positive charges there.

For example, consider a spherical shell with a charge $q$ at its center. This charge will cause a charge of $-q$ to move from the exterior surface of the shell to its interior surface. Therefore there will be a net charge of $-q$ on the inner surface and a net charge of $+q$ on the outer surface. If you calculate the electric fired from this configuration, you will find there is an electric field outside the shell, even though there is no field inside the shell.

May I suggest a simpler case where the same problem seems to arise? A charged conductor with no cavity. How can charge (positive, say) on its surface give rise to a field outside the conductor, but no electric field inside? Surely the surface charge should give rise to an electric field that, inside the conductor, points inwards, and outside the conductor, points outwards, in the $hat n$ direction, say. It does!

The paradox is dissolved by noting that the surface charge is not wholly responsible for the field. The surface charge will induce negative charge on surrounding objects. This induced charge will give rise to an electric field that outside the original charged conductor is in the $hat n$ direction, and, inside the conductor is still in the $hat n$ direction, because the charges that give rise to it are not local to the surface. The charges on the conductor, and the distant charges, distribute themselves in such a way that the two fields are normal to the conductor's surface, cancel inside the conductor but add outside.