From what I have read, outer hair cells in the human ear amplify incoming signals and inner hair cells "pick-up" the signals and generate action potential. However, neurons have refractory periods during which they cannot fire again. Does this mean that the human ear has a "sampling period" within which it cannot "pick-up" sounds?
Inner hair cells (IHC) do not fire action potentials themselves. It's the auditory-nerve that synapses with IHC that generates action potentials. The firing rate of the auditory nerve can be as high as few hundred Hz with a refractory period as short as 1 ms or so (depends on the animal).
However, it is important to note that the signal is not sampled at this rate. As you may know, the auditory signal is first transformed to the "frequency" domain through the physical structure of cochlea. Each inner hair cell therefore is roughly only encoding the relative strength of a frequency band. There are presumably inner hair cells in the human cochlea that correspond to a range centered around 18 kHz for example, but neither the neurotransmitters of the corresponding IHC nor the auditory nerve can fire at 18 kHz. Nevertheless, the amplitude modulation at this high frequency is what is transmitted.
Also, thinking of neural firing as a "sampling period" is not always a good analogy. There are debates about this, but it could be that precise timing of action potential carries large amount of information about the stimulus (perhaps not so much so in early auditory system.)
If you want to see some computational modeling work for inner ear, IHC, and auditory nerve, I recommend the Meddis IHC model: