  We constrain cosmological and galaxy-bias parameters using the combination of
galaxy clustering and galaxy-galaxy lensing measurements from the Dark Energy
Survey Year-3 data. We describe our modeling framework, and choice of scales
analyzed, validating their robustness to theoretical uncertainties in
small-scale clustering by analyzing simulated data. Using a linear galaxy bias
model and redMaGiC galaxy sample, we obtain constraints on the matter density
to be $\Omega_{\rm m} = 0.325^{+0.033}_{-0.034}$. We also implement a
non-linear galaxy bias model to probe smaller scales that includes
parameterization based on hybrid perturbation theory and find that it leads to
a 17% gain in cosmological constraining power. We perform robustness tests of
our methodology pipeline and demonstrate the stability of the constraints to
changes in the theoretical model. Using the redMaGiC galaxy sample as
foreground lens galaxies, we find the galaxy clustering and galaxy-galaxy
lensing measurements to exhibit significant signals akin to de-correlation
between galaxies and mass on large scales, which is not expected in any current
models. This likely systematic measurement error biases our constraints on
galaxy bias and the $S_8$ parameter. We find that a scale-, redshift- and
sky-area-independent phenomenological de-correlation parameter can effectively
capture the impact of this systematic error. We trace the source of this
de-correlation to a color-dependent photometric issue and minimize its impact
on our result by changing the selection criteria of redMaGiC galaxies. Using
this new sample, our constraints on the $S_8$ parameter are consistent with
previous studies, and we find a small shift in the $\Omega_{\rm m}$ constraints
compared to the fiducial redMaGiC sample. We constrain the mean host halo mass
of the redMaGiC galaxies in this new sample to be approximately $1.6 \times
10^{13} M_{\odot}/h$.
