The thesis is devoted to compact object astrophysics and fundamental physics with current and future gravitational wave (GW) observations. Since the first detection of GW event GW150914 by LIGO in 2015, there have been more than 90 transients observed by the LIGO, Virgo and KAGRA collaboration up till now. Additionally, more observations of stellar mass compact binaries would be detected by current and the second generation detectors as well as observations of Extreme Mass Ratio Inspirals (EMRIs) by future space-based gravitational wave detectors such as LISA, TianQin and TaiJi. At the beginning (Chapter 1), a review of data analysis techniques on gravitational waves is introduced, followed by an introduction to the focused projects. In Part II, we propose a new hybrid waveform model, which illustrates comparable accuracy in characterizing tidal effects as the effective-one-body (EOB) waveform and numerical relativity simulation. Part III investigates an important extreme mass ratio inspirals (EMRIs) formation channel driven by active galactic nuclei (AGN) accretion disk. Chapter 6 introduces the expectation of a higher rate of occurrence of EMRIs in AGN for spaceborne gravitational wave detectors; and in Chapter 7, we show that the AGN channel is much more promising to produce mass-gap EMRIs.