Amplitude modulation of unit step signal

Your perfectly single-sideband suppressed-carrier modulated sinusoid certainly has a phase which can be measured. However, what you cannot tell is what the contributions of that measured phase from the audio input and the RF oscillator were.

There is another form of single-sideband modulation, in which not only one sideband but also the carrier component is transmitted. This provides a reference which can be used to synchronize the receive LO to the transmit one - normally done to insure exact tuning, but it would also give you the ability to recover the original audio phase.

It is also quite possible, especially with modern DSP gear, to transmit two separate audio channels, one on each side band. This is commonly called independent sideband modulation (ISB).

Many spread spectrum implementations are DSP based and capable of receiving multiple channels at once - GPS being a good example.

Yes it's possible and done in many applications, but probably not in the way you are thinking. For all practical purposes EM waves are by definition transversal when in the far field plane wave mode and/or in free space which is illustrated below. This is how a large majority of long distance communications operate.

This is not often the case in near-field applications or for Gaussian beam (laser) communications. However this little fact doesn't mean you can also "load" them (we electron pushers prefer the term modulate) in a non-transverse way. The E and H fields are all you have to work with so there's really nothing you can do. Either you have TEM, TM or TE modes of propagation.

You can't really add information the way you are thinking...except, you can accelerate the source! This will induce modulation in the form of a doppler shift. You can then demodulate the doppler shift independently from the the data being transmitted.

But as far as I know that's about the only way you could do what I think you are asking in your question. If I missed your point, then please clarify your question.