AM/SSB modulation and peak envelope detector

A 'normal' AM signal carries amplitude and frequency information. The amplitude can be recovered by rectifying only a sideband component of the AM signal but the modulation frequency is found from the difference between the carrier and this sideband. If the carrier is removed, the 'reference' by which the frequency can be determined is no longer there. The normal practice of inserting a locally generated carrier at the receiver, restores the ability to recover the modulation frequency. It also explains why misplacing the carrier (or the signal relative to it) results in the audio pitch being too high or low.

So the envelope detector when it demodulates an AM signal, works as a simple amplitude detector (to recover sideband amplitude variations to audio amplitude variations) but ALSO as a mixer (to recover sideband frequency variations based on the reference AM carrier back to audio frequency variations), am I right?
This also verifies my thoughts that the sideband in an SSB signal is both amplitude varied and frequency varied.
Am I correct in my above points?

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Yes, that's right. The amplitude of the sideband is similar to the original audio and the sideband does move in frequency. The only thing stopping it being demodulated is a fixed point from which the frequency can be measured from. On it's own, all you see on a spectrum analyzer is the relative positions and amplitudes of the components in the modulation. With voice, that could be quite a complex assortment of frequencies and levels, unless you can talk in sine waves!

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I read somewhere that the carrier in AM, does not only provide the reference frequency for demodulation but also the reference phase and reference amplitude, which is useful for AGC receivers. Are these true? But why the reference phase is needed?

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1. AGC (Automatic Gain Control) adjusts the gain of signal amplifying stages to optimize the levels at the detector (if it's actually possible). It needs the complete AM signal to do that, if the sidebands alone were used, the AGC would track the modulation instead of the signal as a whole. Bear in mind that AGC should work even if the carrier isn't modulated. If you listen to SSB on a HF receiver, you will probably hear the background 'breathing' as the audio level goes up and down, that wouldn't be acceptable in regular AM which could be carrying voice or music.

2. Some receivers (I have one here) use synchronous AM detectors. It uses a PLL to lock to the received carrier frequency and phase as it leaves the IF stages then uses the PLL oscillator as a substitute for the transmitted carrier in a DSB demodulator. It has the advantage over a regular detector of making it immune to selective sideband fading but has the drawback that it needs enough carrier to lock to. It doesn't work at all on very weak signals but gives better quailty if the signal is strong enough.

3. A regular detector picking up an AM signal can be mathematically modelled as a multiplier. The phase of the carrier comes into the equation, without it the two sideband signals may add or subtract, possibly distorting the recovered wave shape. It becomes like an IQ demodulator with two unlocked LO instead of one with a fixed phase shift.

All right, so the phase AND the amplitude AND the frequency of the carrier is needed to correctly demodulate the sideband.