1.) Not quite. The Gross-Pitaevskii equation describes the behavior of a particular wavefunction (i.e. a mathematical object encoding measurable information about an object), not waves in a BEC. In particular, it describes the ground state (i.e. lowest energy configuration) of an ensemble of identical bosons. This, in turn, approximates the state of a BEC at absolute zero. Any finite-temperature BEC (and indeed, any that we can actually create) exists as a superposition between the ground state described by the Gross-Pitaevskii equation and many other excited states, which can probably be obtained through the Hartree-Fock approximation. Anything resembling a wave (i.e. a traveling excitation in BEC) would exist as an excited state of the BEC, so they are not describable using the Gross-Pitaevskii equation. I should probably also add that this equation has little to do with superfluidity.
2.) Since the limits of this equation are determined mainly by the limits of the Hartree-Fock approximation and pseudopotential model used to derive it, I refer you to these sources:
3.) Not sure, as I am not a condensed matter physicist. In any case, the equation arises from a relatively standard approximation in quantum mechanics, so I'd be surprised if there was anything more straightforward. It's essentially just the original Schrodinger equation with an added interaction term. Again, though, this equation has little to do with superfluidity at face value.