Why are springs coiled the way they are? Why not some other shape? Is the shape due to its elasticity or something?
The shape is due to the underlying principle of what a coil-spring actually does.
A spring is originally a long (often metal) wire. When you coil it, you are changing the way the forces can be applied to the spring.
For coil springs, you are making two surfaces that are often flat or have connections, that can be coupled with other systems.
The interesting thing is what is actually happening to the wire when you stretch or compress a coil spring. Because of the helical nature of the winding, when you push/pull on the spring, most of the effort is not going into bending or stretching the wire. It is twisting it.
You can visualize this by thinking of a coil from the side. The twisting forces all the coils closer together or further apart, leading to the motion desired.
You can consider it as a long bar experiencing torsion, allowing it to twist. As long as the bar has a constant cross section this relationship will also be linear, which may also have something to do with the shape.
Essentially, my understanding is that the helical shape allows you to use vertical motions to create twisting in the horizontal planes.
Helical shapes seem to do similar things in other applications. Perhaps someone with a better understanding of the math could give a clear description of why helix shapes can change the nature of motion in general.
(for example, torsional springs actually create a spring that resists twisting, but the spring itself is actually acting like a long bent bar; basically the opposite scenario of a linear helical spring)
Well, metals can bend easily. If a length of metal is in the shape of a coil, then this bending can occur over the entire spring at once. Perhaps minimal temporary deformation at each point prevents permanent deformation. In any case, it maximizes lengthening ability while minimizing "effort" by the spring. I don't know if there are other shapes that would work better, but a cylindrical coil is certainly a simple and effective solution.