Why don't capacitors have a minimum voltage rating?
But if the dielectric is thick enough to handle higher voltages, how can a lower voltage, like 30v, attract a charge on the other side?


Why not? The process is completely linear. There's no "threshold effect". I can use a capacitor rated at 50V to couple a signal that might be only a few &micro;V in amplitude.

Also, it isn't the electric field that causes the electrons to move. It's the displacement of electrons (caused by an external source of voltage) that creates the field in the dielectric.

Suppose you have two capacitors of the same value, but one has 100&times; the dielectric thickness (and therefore 100&times; the area) of the other. If you charge them to the same voltage, they have the same charge &mdash; the same number of electrons have been shifted from one side to the other. Sure, the E field is 100&times; less intense in the one with the thicker dielectric, but there's also 10,000&times; as much (thickness &times; area) of it, so the total energy stored is also the same.