For cinematic content, the commonly used range of focal lengths (for super35, same as APS-C more-or-less) is 18mm through 85mm, with the majority of work tending toward the wide end for most directors and cinematographers. For FF the equivalent range of focal lengths would be 28-135mm.
For under $600, I'd get the Sigma 17-50mm OS f2.8 zoom because this covers most of the useful focal lengths for cinematic production (very generally, some prefer other focal lengths) at a decent speed. f2.8 (to f5.6) is a pretty normal stop for interiors and you'd go deeper for day exteriors (generally, but still stock up on some ND filters to keep your shutter speed around 1/50) so you're all set. That said, you might want faster for night exteriors and some directors (Fincher, for instance) shoot wide open.
But that could be totally wrong. If you're shooting a Ridley Scott style movie you would want to go much tighter. If you're doing an Ozu homage just get the 50mm f1.4 and be done with it.
This depends on genre and style. Each lens has a very different voice and different implications with respect to character identification, subjectivity, etc. It's not THAT important if your content is good (a good script is a thousand times more important), but it can matter. For that reason I'd pick up some primers on cinematography if you're planning on using the short as a launching point for DPing or getting into the technical aspects of filmmaking and reading them through and seeing what your favorite focal lengths are before buying a lens. If you just want to tell a story well... hire a DP or at least a photographer who is very technical. Buy some bright light bulbs (and avoid night exteriors in poorly lit areas) and you can get by on natural light, though... for the most part. Lighting is not as crucial as it once was, though it is very important for getting a specific look. ND filters are essential I've found.