Does the nature of nervous impulses give us a finite number of things we can perceive?
I know we have sensors in the form of hairs that trigger a nervous
  impulse to the brain when they are stimulated. But as I understand it,
  each one can only send that on/off binary signal when they are
  triggered by a very specific level of sound.


Your overall question borders on philosophy and the biology of consciousness which is hardly understood at all. However, your picture of sensory input is a huge oversimplification. That's not really your fault, it's the sort of thing you are likely to pick up from pop science descriptions, which in turn are faulty because this is not a dead field where everything is understood to a T. Even the question of which parts of perception "are" digital and which "are" analog is still under investigation. However it makes your question hard to answer because it's sort of ill-posed since is assumes a purely digital model.

To address one oversimplification, our perceptions of the outside world are not necessarily binary on off bits. It is hypothesized that some neurons signal by changing the rate at which they fire. Is the rate continuously variable? It's not clear, but it certainly provides a much larger range of encodings than 1 and 0.


  Can we only perceive image A and image B with nothing in between, or
  can we perceive image A.012744 and A.064212454 And A.24545157845?


I don't know how this meshes with your statement that 'And I know we all have our personal reactions and even perceptions'. If you've ever looked into color blindness then you already know that some of your fellow humans can't distinguish between A.012744 and A.064212454, even if you can. It's not that they can't see them, they just can't tell that they are different.