The role the lattice plays in superconductivity became clear when the isotope effect was discovered. This is the dependence of critical temperature on the isotope of the superconducting material. Such a dependence is possible only if lattice vibrations are relevant for superconductivity. So if one tries to excite these pairs, one needs a certain energy to break them up. The electron gas is substituted by a gas of Cooper pairs , which are, in a sense, similar to a He atom.
Cooper pairs are bosons, so they all can be in the same quentum state.
The superconducting state is a superfluid state of the Cooper pairs. The energy gap is nothing else but the energy needed to break up Cooper pairs. This will of course destroy th esuperconducting state. The phase of this wavefunction will become very important in the dicussion of the Josephson effect. Single electrons can tunnel through only, because cooper pairs do not exist on the normal side.