We study the effects of particle shape and self-propulsion on the collective behaviors of a two-dimensional granular fluid, using an experimental system of hard square grains. We energize the system by vibration, which, depending on particle shape, induces either isotropic diffusion or persistent self-propulsion in the particles. We use specially designed grains as a model system to study (i) the equilibrium packing of hard squares in two dimensions, (ii) the dynamics of athermal self-propelled particles, and (iii) the melting kinetics of an unconfined granular crystallite. The first study concerns the phase diagram of a two-dimensional fluid of hard squares, which exhibits ordering in the positions and orientations of particles. We measure the structure and dynamics of steady states of vibrated square grains, and identify the progression of phases as a function of packing fraction. At low density, the squares form an isotropic fluid; at intermediate densities, four-fold bond- and molecular orientational order emerge simultaneously to form a tetratic phase with slowed rotational dynamics and short-range translational correlations; at higher densities, the particles freeze into a square-crystalline phase with suppressed translational diffusion, and both translational and orientational order. In vibrated granular active matter, both noise and self-propulsion derive from the same collisional forcing, unlike many other active systems where there is a clean separation between the two. Using the same experimental setup, we study single-particle motion of self-propelled particles. We use a theoretical analysis to compare grain motion at short and long time scales to the assumptions and predictions, respectively, of the active Brownian particle (ABP) model. We demonstrate that despite the unique relation between noise and propulsion, granular media do show the generic features predicted by the ABP model and indicate that this is a valid framework to predict collective phenomena. Finally, we study the kinetics of melting an unconfined crystallite of square grains. We prepare the initial state with an ordered crystalline structure in the positions and orientations of the squares, then suddenly initiate vibration. As crystallites of passive and active particles evolve, we measure the crystallite size, dilation, and orientational order. We find that crystallites of passive particles exhibit ordering remarkably similar to that in the steady-state phases, with bond- and molecular orientational order coupled. Self-propelled particles decouple these; coherent "brick-layer" rearrangements destroy bond-orientational order while molecular orientations remain well ordered. Depending on the arrangement of the self-propulsion axes, activity can either frustrate or promote the melting process, dramatically change the crystallite lifetime, and affect the spatial distribution of the melting front.