A variety of soft and hard condensed matter systems are known to form stripe patterns. Here we use numerical simulations to analyze how such stripe states depin and slide when interacting with a random substrate and with driving in different directions with respect to the orientation of the stripes. Depending on the strength and density of the substrate disorder, we find that there can be pronounced anisotropy in the transport produced by different dynamical flow phases. We also find a disorder-induced "peak effect" similar to that observed for superconducting vortex systems, which is marked by a transition from elastic depinning to a state where the stripe structure fragments or partially disorders at depinning. Under the sudden application of a driving force, we observe pronounced metastability effects similar to those found near the order-disorder transition associated with the peak effect regime for three-dimensional superconducting vortices. The characteristic transient time required for the system to reach a steady state diverges in the region where the flow changes from elastic to disordered. We also find that anisotropy of the flow persists in the presence of thermal disorder when thermally induced particle hopping along the stripes dominates. The thermal effects can wash out the effects of the quenched disorder, leading to a thermally induced stripe state. We map out the dynamical phase diagram for this system, and discuss how our results could be explored in electron liquid crystal systems, type-1.5 superconductors, and pattern-forming colloidal assemblies.