Is it possible to conduct a hypothetical experiment where gravitational wave interference occurs and where areas of constructive and destructive gravitational wave interference are present? If such were possible, would sampling of areas of destructive interference yield a finding of true zero gravity in the universe (unlike Lagrange points)?
Gravitational waves are small perturbations in the spacetime metric caused by specific types of acceleration by masses. The gravitational attraction between masses is always present, regardless of whether they are accelerating and therefore regardless of whether gravitational waves are present.
Hypothetically, you could set up the experiment you propose. You could have two identical oscillating mass quadrupoles (e.g. identical binary star systems), but which were quarter of a phase different in their oscillation phase. These would produce gravitational waves that were exactly out of phase and should "cancel" in the region between the two systems. However, the two binary systems would still be attracted to each other.
At the risk of stepping beyond my area of expertise, the above relies on the assumption that the metric perturbation due to the wave is small, as is always the case for GWs detected at the Earth from astrophysical sources. This allows the Einstein field equations to be reduced to linear second order differential equations in the GW strain (so-called linearised gravity), which means the solutions could be superposed in the way I suggested.