Can signals travel "backwards" in the sensory pathway?
Reverse signals (dendrite -&gt; axon) do occur in neurons, and are called back propagating action potentials (bAPs). However, whatever role bAPs play in the nervous system at large is subtle/small enough that we don't really understand them at all.
In any case, as @luigi points out, pinched nerves don't have anything to do with bAPs. The reason why a pinch in one place (the neck) feels like it has an effect in another place (the arm) is because some neurons trace long (on the order of several feet) pathways through the body. Ultimately, the sensory input of everything below your head passes through neurons in your neck at some point. If there is a pinch in the neuron in your neck that relays sense information from your arm, that neuron in your neck can activate spuriously, even in the absence of any stimulus acting on your arm. In that case your brain will interpret the spurious activation of that neck neuron as being the result of some kind of strange arm stimulus, and so you feel pins and needles.
See here for more information on the sensory cortex in your brain and the &quot;sense image&quot; it maintains of your body.
Edit
To clarify what I mean by a &quot;pathway&quot;,

The part marked &quot;proprioceptors or mechanoreceptors&quot; is what's buried in your skin. If you poke a mechanoreceptor, it evokes a signal in the neuron marked &quot;first-order&quot;. In some cases, that first-order neuron will reach all the way to brainstem (as in the picture above), and in some cases that first-order neuron will interface with an intermediate neuron in the spine, but the effect is the same in any case. A first-order neuron stimulates a second-order neuron, which in turn stimulates a third-order neuron, etc., all the way until the original signal arising from the mechanoreceptor reaches the somatosensory cortex. A pinch in any of those first, second, etc. order neurons will potentially send spurious signals to the somatosensory cortex.
Second edit:
Rereading your question, I think a lot of the trouble that you're having has to do with your understanding of how the brain assembles sense information. In order to feel pain in your arm, it is not necessary for anything to be happening to the arm itself. Your arm could be perfectly healthy and normal, but as long as your brain is receiving a signal equivalent to the one produced by a damaged arm you will still feel the same sensation of pain.