Understanding Gravitational Waves and Their Sources: Robust Inference, Tests of Gravity, and Future Prospects

As gravitational-wave detectors have become increasing more sensitive since the first detection in 2015, the now routine observations of gravitational waves have provided a lens through which the field of gravitational-wave astronomy has been able to study the universe. In this thesis, I explore a substantial number of facets regarding the inference challenges associated with observations from binary compact object mergers. I demonstrate the difficulties conducting and interpreting accurate spin measurements from real observations. In addition, I then present a framework for testing general relativity from an ensemble of events without underlying statistical assumptions. This framework is then extended to incorporate theoretically motivated information into these tests. These methods were utilized to analyze observational data from the LIGO-Virgo-KAGRA Collaboration's third observing period. Additionally, I present a novel summary statistic for diagnosing model misspecification in astrophysical compact binary coalescence population studies. Finally, I conclude with a demonstration of the utility of novel detector readout schemes for future gravitational-wave interferometer designs. My thesis presents a sweeping view of a number of current research avenues with current and future gravitational-wave detectors.