Matter near a gravitational lens galaxy or projected along the line of sight (LOS) can affect strong lensing observables by more than contemporary measurement errors. We simulate lens fields with realistic three-dimensional mass configurations (self-consistently including voids), and then use lens models to quantify biases and uncertainties associated with different ways of treating the lens environment (ENV) and LOS. We identify the combination of mass, projected offset, and redshift that determines the importance of a perturbing galaxy for lensing. Foreground structures have a stronger effect on the lens potential than background structures, due to non-linear effects in the foreground and downweighting in the background. There is dramatic variation in the net strength of ENV/LOS effects across different lens fields; modeling fields individually yields stronger priors on \(H_0\) than ray tracing through N-body simulations. Lens systems in groups tend to have stronger ENV/LOS contributions than non-group lenses. In models, ignoring mass outside the lens yields poor fits and biased results. Adding external shear can account for tidal stretching from galaxies at redshifts \(z \ge z_{\rm lens}\), but it requires corrections for external convergence and cannot reproduce non-linear effects from foreground galaxies. Using the tidal approximation is reasonable for most perturbers as long as non-linear redshift effects are included. Yet even then, the scatter in \(H_0\) is limited by the lens profile degeneracy. Asymmetric image configurations produced by highly elliptical lens galaxies are less sensitive to the lens profile degeneracy, so they offer appealing targets for precision lensing analyses in future surveys like LSST.