The hair cells are not themselves neurons. They're transducers. They're intimately invested by afferent fibers from the neurons in the spiral ganglion. The hair cells are so constructed that any deformation of their microvilli (which is what the "hairs" really are) will cause a change in the overall membrane potential of the cell. In other words, mechanical stimulation causes them to "fire" and this signal is detected by the fibers from the cells in the spiral ganglion. These fibers are neural elements, and they carry their own depolarization wave back to the soma and thence via the efferent fiber into the auditory region of the brain.
But where does the mechanical force to deform the "hairs" come from? That's where the membranes are involved. Relative movement of the organ of Corti (and its hair cells) with respect to the tectorial membrane is the source of the deformation of the "hair cells."
When a sound wave impinges on the eardrum, the lever system of the ear ossicles sends a pulse of energy into the perilymph via the oval window. The "shock wave" set up in the perilymph travels through it to the scala vestibuli; it then moves up the scala vestibuli to the helicotrema. At that point the shock wave begins to move down the spiral via the scala tympani. It eventually reaches the round window and its force is dissipated. All this motion of waves up and down the spiral brings about physical displacement of the membranes separating the perilymph from the endolymph.
This in turn sets up movement in the endolymph. As the basilar membrane (BM) and the vestibular membrane (VM) move they cause the organ of Corti to move, and the hair cells on it are scraped along the bottom of the rigid tectorial membrane. The tectorial membrane doesn't move. It can't: it's firmly anchored to the bone. Banging the hair cells against the tectorial membrane causes the deformation of the hairs and firing of the hair cells.
The shape and nature of the organ of Corti and the cochlear duct are such that different regions respond to different frequencies of sound. In effect, hair cells are "tuned" to different frequencies and can respond to no others. If they die from disease, age, or trauma, the response to that frequency is lost with them. This is a common phenomenon in older animals.
As you might expect, overproduction of endolymph can severely affect the hearing sense. It elevates the pressure in the endolymphatic spaces, causing ringing or buzzing in the ears, and/or dizziness. This condition is known as Meniere's disease and it can arise from a number of causes.
Inner ear; H&E stain, paraffin section, 100x