What would the Aarakocra language sound like

[Ki] is an element of the magic that suffuses the multiverse—specifically, the element that flows through living bodies. (PHB 76)

In the context of the above statement, we should understand the term "spoken" in the Tongue of the Sun and Moon ability description to mean "sound created by a living being". This is reinforced by the claim that the monk learns to "touch the ki of other minds".

1. The monk communicates and receives these meanings by a sort of mental bridge between herself and the other creature. The speech is also occurring for the monk, but she has no automatic comprehension of it. This is also supported by the fact that written language is excluded; a piece of writing has no present mind to communicate it to the monk, and so she cannot read it.
2. No. A recorded audio message suffers from the same limitation as a written message: it is mindless.
3. No. The understanding is conveyed by the monk herself, and if she leaves, then the message is mindless. The spell Magic Mouth requires the monk to "speak the message". The monk is incapable of speaking the message in other languages, and as the spell is not a creature, it cannot understand the mind-speak that the monk uses to communicate to other creatures.

Any bird is hard to prove, but there are certainly variations

The best known variation of Aarakocra tend to resemble eagles.

The ones in the Tomb of Annihilation campaign book, where it describes the Aarakocra of Chult, resemble parrots.

However, both of these are mentioned in the description of Aarakocra already, so might not be the kind of proof you are looking for.

Aarakocra from the Dark Sun setting resemble giant vultures. As there are no vultures mentioned in the description, and they still exist in some setting, there is a strong possibility that there are owl-like aarakocra in other settings.

Historically, even the eagle-like Aarakocra looked different. In earlier editions, they looked more like songbirds. (Thanks, Forrestfire)