Relationship between nerves and axons
I will go through your list of questions below:


I wanted to know how similar is our nervous system to [the circulatory system]?
They are very different, but as in every comparison of very complex systems, there is some overlap. The circulatory system carries fluids, the nervous system electrical signals so they are functionally not alike. However, both systems run throughout the body and have a more or less central control unit (the brain and heart respectively). So there is a structural similarity. I reckon they are as much alike as a city's sewer system and electricity grid.
I understand we have neurons (please correct me if I am wrong) all over the surface of body.
That is correct
Whenever we feel a touch (stimulus arises) a neuron fires a response, and that response travels through axons (myelin sheath).
Tactile stimulation may result in firing of one or more neurons, dependent on the intensity of the stimulus. Larger stimuli will, obviously, recruit more fibers. Stronger stimuli will evoke stronger responses (increased firing). Axons indeed conduct the neuronal responses, like an electrical wire conducts electrical current. Longer axons are often myelinated (insulated). Not all neurons related to touch are myelinated, however. 
So my main question is that what exactly is a nerve?
A nerve is a bundle of axons that carry related functional information. Typically, nerves conduct information originating from locations closely located together in the body, and generally they convey this information to a localized spot in the body as well. For example, the optic nerve carries information from the photoreceptors (related information) from the retina (localized source) to the brainstem (localized target).
Is [a nerve] a long axon?
A nerve contains many axons. 'Long' or 'short' is rather subjective.   
How many axons (same thing as neuron body?) are in a nerve? I am sure it depends on different nerves.
Neuronal cell bodies are typically located outside a nerve. Nerves contain varying numbers of fibers. For example, the auditory nerve harbors the axons of the spiral ganglion cells that transmit auditory information from the inner ear to the brain. It contains between 31k - 32k (myelinated) fibers in normal-hearing humans (Spoendlin &amp; Schrott, 1989). In contrast, the optic nerve, which carries visual information from the retina to the brain, contains some 1.3 million (myelinated) fibers in young human adults (Jonas et al., 1992)




References
- Jonas et al., IOVS 1992;33:2012-8
- Spoendlin &amp; Schrott, Hear Res 1989;43:25-38