A vibrating shoelace is a poor analogy for a sound wave, because that would be a transverse wave whereas sound is a longitudinal wave.
What this means is your example of molecules hitting each other is perfect. A longitudinal wave is described by the variation of the density of the particles. The Wikipedia page has some nice animations to help visualize what that means. The wavelength in this context is the distance between two periodic fluctuations in the density of the material.
So in the case of the question you linked to, because the sound emitted has a massive wavelength (and consequently an extremely low frequency) that means the distance between the density fluctuations are large. So the particle literally moves and collides with another particle and so on, propagating the wave.
We can consider four aspects of your question:
- Why do most events generate sound?
- What sounds get propagated?
- What does it take for sound to be detected?
- Has evolution got anything to do with this?
1 - generating sound
Most of the sounds you describe are "broad band". Remember that a delta pulse (short sharp shock) is basically "all frequencies", although in reality a pulse of finite duration will not contain the very highest frequencies. Now it turns out (see for example my earlier answer on this topic) that it takes an absolutely TINY motion (less than an atom's width) to generate an audible sound pulse - so we can safely say "every motion makes a sound; most motions make audible sound".
2 - propagation of sound
Like all finite-sized sources of energy, once you are a reasonable distance (reasonable compared to the size of the object generating the sound) away, sound intensity falls off as the inverse square of the distance (barring mechanisms to contain the direction of propagation: tunnels, mountains etc). This means that sound will typically remain audible for roughly the same distance as the object making it remains visible/interesting. Certain very loud sources (e.g. crickets) are an exception to this rule - but they are deliberately trying to be heard a long way off (see point 4). Sound is also attenuated by air - according to Stokes's Law, the attenuation coefficient $\alpha \propto \omega^2$, meaning that higher frequencies are absorbed more strongly (because of viscous interactions in the air). From the Bruell & Kjaer website:
Low frequencies really only get attenuated according to the inverse square law, but higher frequencies are attenuated more strongly.
3 - detecting sound
In order to detect sound, a membrane needs to be moved. This motion then has to somehow be conveyed to the nervous system, which is water-based and therefore has a very different acoustic impedance than air ($z_0 = \rho c$ - so when density increases by 1000x and speed of sound by 4x, you have a mismatch...). The mechanisms in the ear (tympanic membrane, malleus, incus, stapes, oval window, cochlea) is a beautiful piece of engineering to create something of an acoustic match, and works quite well over a range of frequencies. Unfortunately, for very low or very high frequencies, bit of that mechanism stop working so well - the finite mass (inertia) of the components makes them more reluctant to move at high frequencies. This again puts an upper limit on the frequency we can hear. However, the "amplification" that the entire organ provides is exquisite - as I computed in the answer linked above this means you can hear tiny, tiny vibrations.
4 - evolution
The human body is a wonderful machine, refined by aeons of evolution - "she who hears the approaching predator lives to procreate another day". The combination of "everything disturbs the air around it" and "we are designed to detect the slightest sound" is the answer to your question.
Best Answer
Sound generated aerodynamically is a mature topic. There are some relevant lecture notes here: http://people.bath.ac.uk/ensmjc/Notes/tnoise.pdf (Although if you have access it's worth your while to read Lighthill's 1952 account).
In essence, correlations of Reynolds stresses (recall a stress is just a flux of momentum) due to regular fluctuations or turbulence in the flow can be shown to be equivalent to a distribution of acoustic quadrapoles, hence generating sound.
If you want a heuristic understanding then it's worth going through the thought experiment of sound generation due to a pulsating sphere. The velocity field of the fluid is related to the pressure through the Navier-Stokes equations. It can then be shown that fluctuations of the momentum imparted by the pulsating sphere to the fluid drives a pressure field and therefore sound is created.