Can you "rewind" sound messages by going supersonic?

Isn't it wonderful? Hearing the same message again and again without any effort from your part!

No it is not possible, obviously, because energy cannot come from nowhere. We did not even need to think about the specific details in your question.

Your friend yells out something at you, but you can't hear it, because it's Mach 1.

You cannot hear it, but not because of what you think. At Mach 1, everything your friend does (including moving his/her vocal chords) will result in little sonic booms around him/her. The vibrations cannot propagate faster (in terms of group velocity) than the speed of sound in air, which implies that you will not hear it. Also, those wavefronts will dissipate if the plane remains exactly at the speed of sound.

You ask the pilot to slow down the plane, and the plane gradually decelerates.

Your friend's sound catches up with you, and you hear the message!

Totally false! You will hear none of what your friend said during the time the plane was at Mach 1, and merely hear some of the sonic booms that had not yet dissipated, because all their wavefronts will simply accumulate just in front of your friend once the pilot starts decelerating. You will start hearing only what your friend says after the plane has started decelerating, and you will hear it at roughly the same rate it is spoken, but shifted slightly according to the net change in plane's velocity between the time it left your friend's mouth and the time it reaches your ear. [This clarification is in response to a comment. I had treated this effect as small since the deceleration was stated to be gradual.]

Actually, via simple reasoning it should have been obvious that you will not hear most of what your friend said before the deceleration, because otherwise your friend could speak an arbitrarily long message, but it is impossible for you to hear an arbitrarily long message in a finite span of time (the plane can decelerate to zero velocity in finite time).

Now, you yell out something at your friend facing forward, realising that if you had faced backwards, your message would get ultrasonic for your friend.

This makes no sense. Whether you are facing forward or not, the vibrations in the air that you produce will propagate outwards in roughly the same manner if the plane is travelling at less than Mach 1. The speed of propagation is the speed of sound in air, so it has nothing to do with the direction you are facing. You are not throwing a ball. You are making vibrations in the air.

As explained earlier, if you are facing backward, your friend hears it at the same rate it was spoken. If you are facing forward, you friend will also hear it similarly just muffled because you are blocking.

Now, you speed up the plane, and your friend catches up with the message. Now, something remarkable happens when you do this. Your friend hears the message in reverse!! If you keep on accelerating and decelerating the plane about the speed of sound, you can make your message to be heard multiple times by your friend!!

Even if you look at the 'forward-facing' part of the spherical wavefront, so that you friend can catch up with it later, note that the longer you wait the further the distance it has to travel and the lower the intensity. Also, once your friend catches up with it, it 'breaks' around him/her and will be gone, so he/she will not hear it multiple times.

In sum, your friend will hear what you say the first time from the 'backward-facing' part of the wavefront, which should arrive no matter whether you face forward or backward, and will hear what you say when the plane is at less than Mach 1, rather softly and in reverse the second time from the 'forward-facing' part of the wavefront, but only if you are facing forward when you said it. Also, he/she will not hear anything you say facing forward while at Mach 1 or higher, because the wavefront will 'break' against yourself and dissipate.

UPDATE: Most of the comments posted here are about the rewinding part. In fact, that is only a part of the question. The real question emphasizes on the idea, of receiving a signal by overtaking it. The energy loss due to the sound propagation can be easily solved by using a hollow straight tube, between the friends.

If you tightly seal one end of your tube to the mouth of the source and the other end to the ear of the target, then clearly you have a stationary medium relative to them through which sound will travel normally. You are effectively putting their mouth and ear back inside the plane, where as you know you can yell from the front to the back with no problem.

If there is a gap at both ends, obviously if the gap is not too small the air will flow through the tube at roughly the same velocity as outside the tube, only slightly lower. So the effects will be pretty much the same as having no tube. In particular, the 'forward-facing' wavefront when you are at less than Mach 1 travels forward, not backward into the tube, so it will not help your friend hear it.

The possibility of receiving a message or music in reverse was discussed by Lord Rayleigh, in his Theory of Sound (1896). If the source is moving toward the observer at twice the speed of sound, a musical piece emitted by that source would be heard in correct time and tune, but backwards.

Well, there is actually a lot of effort on your part (going mach 1 outside a plane is not easy). There are two bigger problems though. The intensity of the sound is decreasing with distance as 1/d^2. So when you 'hear' your friend it is like you trying to hear him when he is a few hundred metres away. Apart from trying to detect such a low signal, there is also a supersonic boom, making it even worse.

EDIT: Since your question has also an edit: If you put a tube between the two friends, it is true you dont have a problem with sound volumen, but then you also dont have any of the effects you are looking for. This would be the same like sitting inside the airplane (since the air for communication is stationary between the two friend). Nothing weird happening inside a concorde.