The reason why you see Fourier transformation applied two times in the feature extraction process is that the features are based on a concept called cepstrum. Cepstrum is a play on the word spectrum - essentially the idea is to transform a signal to frequency domain by Fourier transform, and then perform another transform as if the frequency spectrum was a signal.
While frequency spectrum describes the amplitude and phase of each frequency band, cepstrum characterizes variations between the frequency bands. Features derived from cepstrum are found to better describe speech than features taken directly from the frequency spectrum.
There are a couple of slightly different definitions. Originally cepstrum transform was defined as Fourier transform -> complex logarithm -> Fourier transform . Another definition is Fourier transform -> complex logarithm -> inverse Fourier transform . The motivation for the latter definition is in its ability to separate convolved signals (human speech is often modelled as the convolution of an excitation and a vocal tract).
A popular choice that has been found to perform well in speech recognition systems is to apply a non-linear filter bank in frequency domain (the mel binning you're referring to) . The particular algorithm is defined as Fourier transform -> square of magnitude -> mel filter bank -> real logarithm -> discrete cosine transform.
Here DCT can be selected as the second transform, because for real-valued input, the real part of the DFT is a kind of DCT. The reason why DCT is preferred is that the output is approximately decorrelated. Decorrelated features can be modelled efficiently as a Gaussian distribution with a diagonal covariance matrix.
 Bogert, B., Healy, M., and Tukey, J. (1963). The Quefrency Alanysis of Time Series for Echoes: Cepstrum, Pseudo-Autocovariance, Cross-Cepstrum and Saphe Cracking. In Proceedings of the Symposium on Time Series Analysis, p. 209-243.
 Oppenheim, A., and Schafer, R. (1968). Homomorphic Analysis of Speech. In IEEE Transactions on Audio and Electroacoustics 16, p. 221-226.
 Davis, S., and Mermelstein, P. (1980). Comparison of Parametric Representations for Monosyllabic Word Recognition in Continuously Spoken Sentences. In IEEE Transactions on Acoustics, Speech and Signal Processing 28, p. 357-366.