Gravitational waves are the best ways to probe any spacetime anomalies or structure, for sizes where general relativity ((GR) applies. That means sizes significantly larger than the Planck,length, where it won't apply. We don't know what spacetime will mean at those scales, and we think it won't be continuous nor identifiable as space time. Maybe it'll be strings, or loops or foamy, we don't know. A gravitational wave at those scales might just be no different than the other fluctuations happening. There is no spacetime.
But at cosmological scales or other macroscopic scales, gravitational waves could see large changes. In fact large changes are likely to emit gravitational waves - for instance large, so called cosmic strings, if they exist, would be likely strong emitters of gravitational radiation. See the wiki article on cosmic strings. https://en.m.wikipedia.org/wiki/Cosmic_string
In fact the simplest effects of spacetime composition, at macroscopic cosmological scales, is the cumulative Wolfe Sachs effect which accumulates redshifts from spacetime voids. This is observed in the CMB (yes, electromagnetic, not gravitationa waves). The same could be observed by gravitational waves (GWs), but we need bigger GW observatories (interferometers) that come closer to matching the wavelengths involved which will be very large.
LIGO has 3 Kms legs, is able to see a few hundred Kms wavelengths (maybe more). A longer leg interferometer would be that much more sensitive and be able to see longer wavelengths from bigger objects. LISA or eLISA, space based and planned for the 2020's, would have a million Kms legs. LIGO and similar and better but earth based interferometers will be able to see inside neutron stars, supernova, and other stellar type objects, and map out their insides some (where electromagnetic waves cannot get through, and see for instance the quark cores of some neutron stars, and other features inside.
As for cosmological features, LISA/eLISA would be able to see some that electromagnetic waves cannot. That would include those cosmic strings and other large features, anomalies or discontinuities.
As far as calculational capabilities, yes we need them to be able to know what to look for. Numerical techniques are used to calculate in the fully nonlinear regime of GR (general relativity), where the gravitational field is too strong and can't be approximated well. That happened, for instance, in the Black Holes (BHs) merging where they got close to each other's horizons, and the full nonlinear numerical calculations. Before it got to that regime they were able to do what is called a PPN (parametrized post Newtonian) approximation, with various levels beyond the Newtonian approximation (they went to 2.5 or 3). As the gravitational field gets stronger it will need the numerical calculations. As @Countto10 implied, the question is whether those can be done for more complex problems.
For the BHs merging the main problem was finding a way to do numerical calculations at different time scales, as time went slower near the horizon, and it needed more steps to do. That was solved about 10 or more years ago, by constructing a different coordinate frame as it got closer, and actually extending it inside the horizon without a coordinate singularity. The calculations are done and are able to compute a large number of posible waveforms that could be generated (the GWs), so they'd know what to look for and use it to do matched filter detection, and parameter estimation to be able to get the BH parameters. A similar type of calculation, with different coordinate systems and issues will need to be done for other cases, but there is no clear showstopper. A summary and description of the methods used in numerical relativity can be read at