In Cooper pair, is there an essential condition on spin of electrons forming pair? Also explain the reason behind it. And what is the charge on them.
The quantum state of electrons forming Cooper pairs are plane wave (free electrons from the Fermi sea of the conducting band). So the spatial part of the quantum state of the pair is symmetric under exchange of the two particles. The total state shall be anti-symmetric (Fermi rule), so the angular momentum part shall be anti-symmetric. But then since neither electron has an orbital momentum, and assuming the pair has no orbital momentum, we are just adding two spin 1/2. The result is textbook: we have an anti-symmetric state with spin 0 and three symmetric states of spin 1. So the spin shall be 0.
As for the charge, the answer is straightforward: -2e