Your ear is an effective Fourier transformer.
An ear contains many small hair cells. The hair cells differ in length, tension, and thickness, and therefore respond to different frequencies. Different hair cells are mechanically linked to ion channels in different neurons, so different neurons in the brain get activated depending on the Fourier transform of the sound you're hearing.
A piano is a Fourier analyzer for a similar reason.
A prism or diffraction grating would be a Fourier analyzer for light. It spreads out light of different frequencies, allowing us to analyze how much of each frequency is present in a given source.
In vauum, all the electromagnetic waves have the same speed $c$. When the wave passes through a material, such as the glass of a prism, the speed is decreased, but only during the passage, inside the material. When the wave exits from the prism, its speed comes back to the speed $c$ (or slightly less, if the wave is propagating in air instead of vacuum).
When the wave enters into the prism, it keeps its frequency (how many periods per second). But, since it is slowed down, its wave length becomes shorter: in a period, it travels a shorter path. But, also in this case, when the wave exits, it takes back its wave length, the same it had when it entered.
Coming to the second question. It is useful to think that the color of the light is associated to the frequency, which never changes in refraction, reflection and diffraction processes. Frequency is important because the "sensors" in our eye are sensitive to the photon energy, which depends on the frequency. On the other hand, the speed of light in our eye depends on the material of the eye itself, so there is also a well defined and fixed relation between wave length and frequency!
So said, a beam with a given color, after passing through a glass prism or a lens, will still have the same color. Of course, its wave length will change along its path, but our eye will never know it.
Finally, what does it mean that a beam is the superposition of various wave lengths? This would deserve a separate question, but this can help to understand the principles:
https://demonstrations.wolfram.com/SuperpositionOfWaves/
Best Answer
There's a bit of a misconception here. A prism causes dispersion, which is the decomposition of a broad spectrum of light into its spectral components via the components' deviation angle from their original trajectory - but this is not in any way related to Fourier transform, rather it's because of Snell's law of refraction, and the fact that refraction changes with the frequency of the light (color).
You can talk about the Fourier transformation of light, but in a different context: spatial frequencies. Much like every sound signal is composed of temporal frequencies, every optical image is composed of spatial frequencies, and one can analyze the Fourier transform of an image to learn about the spatial composition.
One of the most useful cases of Fourier transform in optics is taking the Fourier transform of an optical system's impulse response, which is the image of a perfect point source of light, a.k.a the point spread function (which analogous to linear system's impulse response in signal processing). The real part of the normalized Fourier transform of the point spread function is called the modulation transfer function and is one of the most common metrics to evaluate the quality of an optical system.