A conductor is defined as it has free electrons to move, but when non conductor attain charge, it would also have free electrons. so it should behave like a conductor or not?
When you add excess charge to a non-conductor that charge is not necessarily in a conduction band of the type we get in metals. Typically the extra charge will reside in surface states at the surface of the non-conductor. These are localised states and do not form a band, so the electrons in them are not free to move.
According to electric resistance we (roughly) divide materials in three groups: conductors, semiconductors and insulators.
Some additional knowledge: Fermi level is total chemical potential for electrons, amount of thermodynamic work required to add an electron to the body. In case of conductors, semiconductors and insulators, Fermi level is an energy level of specific material where electrons have 50% probability of being present.
Conduction depends on availability of unfilled electric states. This means that conduction band is the only place where electrons can accelerate due to applied EM field. Conductance is thus ability of electron flow from valence to conduction band.
Conductors have no band gap; this is gap between conduction band and valence band. Lack of band gap causes little or even none resistance.
Semiconductors have small band gap which electrons can breach only if we add some extra energy. By doing so, electrons can "jump" from valence into conduction band.
Insulators, on the other hand, have large band gap. If you wish for electrons to arrive at conduction band you would require tremendous amount of energy. If you manage to achieve this you've caused electrical breakdown of an insulator.
When this phenomenon occurs your insulator will behave more like conductor than insulator. But again, that doesn't mean that such material will completely imitate conductors.
Hope I've given you an adequate answer.
Yes, you can create the charge-rich area in band insulator and make this area behave in every way similar to the band conductor. Probably the most direct example of this is the field effect in semiconductors.
Current is the directional movement of particles, i.e. you should have more particles moving forwards, than backward. Electrons in solids has certain energy-momentum dependence $\epsilon(k)$. In equilibrium amount of electrons with $+k$ and $-k$ is equal and $\epsilon(-k) = \epsilon(+k)$. To create directional movement you need to increase e.g. the number of electrons with $+k$ compared to $-k$. This will require increase in energy for $+k$ electrons.
In band insulators the band is fully filled and there are no available levels in the close proximity of $\epsilon(k)$, so conductivity is not possible at low electric fields. In band conductors there are such levels and conductivity is possible at arbitrary small field.