The primary selection criteria for choosing FM vs. AM is dynamic range

The government decides the frequencies. In the United States, it's decided by the FCC in coordination with the ITU, which is the international governing body (since radio waves don't magically stop at the borders of countries).

There's nothing particular about FM that makes it more suited to those frequencies compared to others. In fact, ham radio operators and others use FM on many different frequency bands. Probably when FM broadcasting needed some spectrum, the 88 - 108 MHz was the most convenient at the time, for technological or political reasons.

Without going in too much detail, just for basic understanding:

The simplest FM demodulator would project the RF signal on the slope of a low pass filter. When the frequency increases the amplitude of the LPF output will reduce, whereas when the frequency decreases the output of the LPF would increase, resulting in an AM signal which conveniently you already know how to demodulate.

A practical FM demodulater would probably consist of a band filter to discriminate the signal you are interested in from all the noise. Then the RF signal is usually mixed down to an intermediate frequency, this improves on the practical component values while being able to tune on a signal from a wide band. Then the above mentioned LPF to demodulate the audio signal.

Modern FM receivers "try to grab on to the IF signal" with a local oscillator. It tries to keep its local oscillator in sync with the IF signal. The control signal for the local oscillator closely resembles the audio signal. This is what is commonly called a PLL receiver (Phase Locked Loop).