Importance of Sine Waves And Maxwell's Equations
If you take the Maxwell equations, they can be combined to derive the wave equation for light.  Since the orthogonal basis for the solution space of the wave equation is the complex exponentials, we can see that sine waves are intimately related to the Maxwell equations here.  When wave propagation is defined by the Maxwell equations / the wave equation, they always must be able to expressed as linear combinations of complex exponential.  This is really the reason why sine waves don't disperse.  Any other wave forms contain superpositions of the sine waves that travel at different speeds, causing dispersion.
What's said here isn't much different from what you've already put together, but it is definitely a fundamental physical reason for sine wave popularity.
Regarding the fantastical third world:
I think that sine waves would still be important because the solutions for linear third order differential equations are still exponentials.  That said, it would also be important to consider (real) exponentials, which are sort of the other extremely fundamental function that is seen everywhere.  Depending on the third order differential equation, there could also be exponential solutions, which could be a very important part of the fantastical world.