Why is the E-field inside a conductor 0?

It doesn't matter whether there's an external E-field or not. The physical reason the E-field inside a conductor is zero, is that in an ideal conductor, all the charges are perfectly free to move around. So if there WERE an E-field inside the conductor, that would cause the charges to move. Which means, they continue to move until they reach a configuration in which the the E-field cancels out to zero, and then they stop moving. They simply adjust themselves until the internal field is zero. This is true whether there's an external field or not.