  The ever-increasing number of detections of gravitational waves (GWs) from
compact binaries by the Advanced LIGO and Advanced Virgo detectors allows us to
perform ever-more sensitive tests of general relativity (GR) in the dynamical
and strong-field regime of gravity. We perform a suite of tests of GR using the
compact binary signals observed during the second half of the third observing
run of those detectors. We restrict our analysis to the 15 confident signals
that have false alarm rates $\leq 10^{-3}\, {\rm yr}^{-1}$. In addition to
signals consistent with binary black hole (BH) mergers, the new events include
GW200115_042309, a signal consistent with a neutron star--BH merger. We find
the residual power, after subtracting the best fit waveform from the data for
each event, to be consistent with the detector noise. Additionally, we find all
the post-Newtonian deformation coefficients to be consistent with the
predictions from GR, with an improvement by a factor of ~2 in the -1PN
parameter. We also find that the spin-induced quadrupole moments of the binary
BH constituents are consistent with those of Kerr BHs in GR. We find no
evidence for dispersion of GWs, non-GR modes of polarization, or post-merger
echoes in the events that were analyzed. We update the bound on the mass of the
graviton, at 90% credibility, to $m_g \leq 1.27 \times 10^{-23}
\mathrm{eV}/c^2$. The final mass and final spin as inferred from the pre-merger
and post-merger parts of the waveform are consistent with each other. The
studies of the properties of the remnant BHs, including deviations of the
quasi-normal mode frequencies and damping times, show consistency with the
predictions of GR. In addition to considering signals individually, we also
combine results from the catalog of GW signals to calculate more precise
population constraints. We find no evidence in support of physics beyond GR.
