Fundamental role of the Fourier Transform

The Fourier transform is, in some sense fundamental. To grasp this, note that you can do Fourier analysis on locally compact abelian (LCA) groups (e.g. as $\mathbb{R}^d$, $\mathbb{Z}^d$ or $[0,1]^d$ (see as a torus)). The central theorem which describes the Fourier transform, is Pontryagin's duality theorem which says that every LCA group is isomorphic to its dual group and the isomorphism is given by the Fourier transform. In the case of $\mathbb{R}^d$ this gives you precisely the well known Fourier transform. If you consider $[0,1]^d$ then this leads to the theory of Fourier series.

A good reference is Rudin's "Fourier Analysis on Groups".

The Fourier transform is just a special case of the Bilateral Laplace transform. The Bilateral Laplace transform is a function of a complex variable s, and if you restrict it only to the imaginary axis, then you obtain its Fourier transform. Example: if $F(s)$ is the Laplace transform of $f(t)$ then $F(i\omega)$ is its Fourier transform.

My understanding of the connection between linearity and the laplace transform( and therefore the fourier transform) is that whenever you have a linear operator which is also time invariant ( or space invariant as the case maybe) such as the derivative operator, then its eigenvectors/eigenfunctions are complex exponentials and in general we would like to represent our vectors/functions in the eigenbasis, the transformation that does that in this case is of course the laplace transform.