Using the first three years of data from the Dark Energy Survey (DES), we use ratios of small-scale galaxy-galaxy lensing measurements around the same lens sample to constrain source redshift uncertainties, intrinsic alignments and other systematics or nuisance parameters of our model. Instead of using a simple geometric approach for the ratios as has been done in the past, we use the full modeling of the galaxy-galaxy lensing measurements, including the corresponding integration over the power spectrum and the contributions from intrinsic alignments and lens magnification. We perform extensive testing of the small-scale shear-ratio (SR) modeling by studying the impact of different effects such as the inclusion of baryonic physics, nonlinear biasing, halo occupation distribution descriptions and lens magnification, among others, and using realistic N-body simulations of the DES data. We validate the robustness of our constraints in the data by using two independent lens samples with different galaxy properties, and by deriving constraints using the corresponding large-scale ratios for which the modeling is simpler. The results applied to the DES Y3 data demonstrate how the ratios provide significant improvements in constraining power for several nuisance parameters in our model, especially on source redshift calibration and intrinsic alignments. For source redshifts, SR improves the constraints from the prior by up to 38% in some redshift bins. Such improvements, and especially the constraints it provides on intrinsic alignments, translate to tighter cosmological constraints when shear ratios are combined with cosmic shear and other 2pt functions. In particular, for the DES Y3 data, SR improves S-8 constraints from cosmic shear by up to 31%, and for the full combination of probes (3 x 2pt) by up to 10%. The shear ratios presented in this work are used as an additional likelihood for cosmic shear, 2 x 2pt and the full 3 x 2pt in the fiducial DES Y3 cosmological analysis.