Still is there electric field around it?
Yes. The electron is moving (in our reference frame), so now there is a magnetic field (in our reference frame), but nothing happens to the electric field.
i. If it has electric field around it, why is it that when electrons are moving in a conductor (i.e.. current if flowing in a conductor) there is no electric field outside the conductor?
The electrons in the conductor produce an electric field outside the conductor; however, realistically, there will be just as many protons in the conductor as electrons, and hence the net electric field outside the conductor is zero.
ii. Now, when a current is flowing in a conductor (I'm not sure what happens if the motion is not inside conductor) it produces magnetic field around it. I'm lost. What happened to the electric field? Is it still there? Are there both electric field & magnetic field? Why don't we discuss about it?
The electric field is still there (in some sense), but its zero, because the electrons and protons in the conductor cancel each other out, so we don't care about it. (Actually, I believe that if you take into account relatavistic effects, which is probably silly not to do in the context of electrodynamics, there will be a nonzero electric field). That being said, if for some reason there were stream of moving electrons with no protons, then we would observe both a (nonzero) magnetic and electric field.
- Hypothetically, If the electron is moving with speed more than that of light. What happens now?
Special relativity says this can't happen :). In any case, if we play dumb for a moment, the only thing that would change is the current, and hence the strength of the magnetic field.