The conduction electrons in metals is a thermal phenomenon? When applying an external electric field in a metal at absolute zero, there is electrical current?
There must be thermal fluctuations in the electron's band to be occurs current?
 A: In a metal the Fermi energy is somewhere in an unfilled band. At any temperature above absolute zero (which you can never reach) there are states available for electrons to get to and result in conduction at the Fermi surface. This will occur in any metal. Superconductivity is a separate phenomena that I won't touch on here.  
A: At present, there is a belief (though obviously not verifiable) by solid-state physicists that a metal cannot exist at absolute zero. The Fermi surface of the metal will be unstable to order of some sort such as superconductivity, charge density waves, magnetic ordering, etc.
With that said, let us concentrate on your scenario though. If there are no phonons to scatter the electrons at zero temperature and the sample is extremely pure (very few defects, impurities, etc.), then one would actually observe the phenomenon referred to as Bloch oscillations. This is when you apply a DC electric field to a metal and you observe an AC current response due to the absence of scattering.
However, in the presence of impurities, I don't see why there couldn't be electrical conduction. There would still be a scattering mechanism, and the applied electric field would still be capable of causing the electrons near the Fermi surface to transition to a state right above the Fermi level.
A: Have you heard of superconductivity? This is a phenomenon where a material exhibits zero resistivity near absolute zero: it clearly contradicts your assertion that thermal excitation is needed for conductivity near absolute zero.
For a semiconductor, it is true that electrons need to be kicked into the conduction band by thermal fluctuations - but for a conductor, the electrons are already there - and they will move in response to an electric field at any temperature. 
A: After going through number of articles for temperature reaching absolute zero, I find that it is difficult to attain absolute zero, which may mean that it is very difficult to stop interatomic movements or energy exchanges, and thus absolute zero is near theroretical. As far as superconductivity is concerned it must be a critical point, minimum energy transfer need and may reach no energy transfer at absolute zero temperature.
