As for the disagreement between GR and quantum mechanics, this is a long-standing theoretical issue. That is, you don't need observations to tell you that GR and QM don't work together; they are incompatible even at a theoretical level. Observations of gravitational waves, or lack thereof, are pretty much irrelevant to that debate. What this means in practical terms is that like classical mechanics GR is a useful approximation in the non-quantum regime, but it breaks down when you get into regimes where quantum effects are important. Physicists have been trying to figure out a new, quantum mechanics-compatible theory of gravity for decades, so it's not like they're waiting for the results of this experiment to come in before they get started on it. However, it turns out that coming up with a good, testable quantum theory of gravity is rather harder than it looks.
The real significance of these gravitational wave observatories is not in theoretical development of quantum gravity, but rather in their role as a new window on astrophysical phenomena. When astronomers began to observe in the radio we learned a lot about known astrophysical pheonomena, and we discovered entirely new phenomena that we never suspected. It seems reasonable to expect similar discoveries from gravitational wave astronomy when it becomes a reality. For example, once ground based detectors develop sufficient sensitivity, some astronomers hope to learn a lot about the makeup of neutron stars from the gravitational wave signature of binary neutron star coalescence. And the LISA experiment, if it ever flies, will have sensitivity in the frequencies that contain the relict gravitational radiation from the Big Bang. Just as the Cosmic Microwave Background has taught us a lot about the early universe, we hope to make similar discoveries from the Cosmic Gravitational Wave Background.