Particle-laden turbulent shear flows are ubiquitous in environmental and industrial flow-systems, and their analysis is thus of prime importance. In this work, we study the motion of a dilute suspension of particles in a non-stationary homogeneous turbulent shear flow (HTSF), subject to varying levels of imposed mean-shear, gravity, and inertia. We use direct numerical simulations (DNS) of the fluid velocity field (coupled with Lagrangian particle tracking), to assess the influence of flow-anisotropy and gravity on the motion of the particles. We first discuss numerical challenges encountered while performing DNS of HTSF at higher Reynolds numbers. The presence of sharp velocity gradients in the HTSF flow field is found to cause premature loss of resolution at the small scales, leading to shortened simulation-times. To counter this, the existing pseudo-spectral DNS setup is augmented with a Weighted Essentially Non-Oscillatory (WENO) scheme, enabling numerically-stable HTSF simulations at higher Reynolds numbers. We then consider the motion of individual particles as they interact with the anisotropic topology of the turbulence. In contrast to isotropic turbulence, particles are found to collect within vortex layers, regions where strong vorticity and strain are coupled with low streamline curvature. Shear-induced anisotropy in the turbulence also leads to reduced gravitational settling speeds for intermediate-inertia particles, though stronger gravity overcomes this effect. Particle velocity variances are found to be highly anisotropic at stronger shear, while gravity now tends to diminish this effect by limiting the interaction-time between particles and turbulence. Shear and gravity acting together cause particle acceleration variances to exceed those of the underlying fluid, corroborating findings from past turbulent boundary layer experiments. Analytical expressions are derived for the mean velocities and accelerations of the particles, and are in agreement with the DNS results. Finally, we analyze the relative velocities and clustering characteristics of particle pairs, and find that stronger shear and gravity suppress path-history effects for particles with stronger inertia. Shear-induced anisotropy in pair-statistics is affected by both inertia and gravity, with stronger gravity seen to oppose the action of shear. Changes in the relative-velocity anisotropy are correlated to the trends shown by single-particle velocity variances. Particle collision rates increase with stronger shear, and are found to scale in proportion to the underlying turbulence timescales.