Energy of Cooper pairs Which is the energy of one Cooper pair? Is it below some energy bands (1s,2s) since a Cooper pair is a boson ?Which is the ground state of a boson (electron -phonon-electron)?
 A: The binding energy of a Cooper pair is the energy of the gap that exists in the superconducting state.  Since the size of the gap is temperature dependent, so is the binding energy.  An approximate formula for the temperature dependence of the energy gap is:  $$\Delta(T)=3.2kT_c(1-\frac{T}{T_c})^{0.5}$$ where $T_c$ is the superconducting transition temperature.
Since Cooper pairs involve electrons in the conduction band, that is where the gap is formed.  The lowering of energy of a Cooper pair does not push their energy below that of core levels of the ions that constitute the lattice.
Typical values for the energy gap at T=0K are in units of $10^{-4}$eV.  For example, aluminum is 3.4 x $10^{-4}$eV.  This is far smaller than the binding energy of core electrons in the ions.
Be careful too!  A Cooper pair is not really a Boson.  I know that the total spin of a Cooper pair is 0, but that is not all that goes into defining what type of particle it is.  There is also how is responds under exchange of particles.  Eisberg and Resnick make this clear in their book Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles:

It is tempting to think of a Cooper pair as acting like a boson, since it contains two fermions.  If this could be done, superconductivity would simply be another example of Bose condensation, as in superfluidity of liquid helium. ...The reason why it is not valid is that the individual electrons in each pair are weakly bound to the pair, which also means the pair is large. As a consequence, the eigenfunction for the system of overlapping pairs must take into account the exchange label of one electron from one pair and one electron from another pair, as well as the exchange of labels of one complete pair and another complete pair.  In the latter exchange the system eigenfunction will not change sign because the two fermion labels are being exchanged, but in the former the eigenfunction does change sign since only one fermion label is being exchanged.  So Cooper pairs are neither purely bosonlike, nor purely fermionlike with respect to all eigenfunction label exchanges that must be considered.

With regard to what an Energy Gap actually is, the following figure may help:

