Thieves' Cant isn't a written language, thus there would be nothing to understand via a spell.

Nowhere in the quote you've pulled (or the PHB) is thieves' cant ever described as a written language. This is because thieves' cant is both verbal and physical communication. Some word substitution (1 to 1) is used, but it is largely based on metaphor and contextual meaning and a big part of this is the hand symbols used when speaking. D&D's basis for thieves' cant is both historical and a trope.

The symbols mentioned are more like pictographic signs than words.

As such they are not translated, but identified, similar to how we use symbols such as the biohazard sign and nuclear sign to signify specific danger or how the symbols on a crosswalk signify when to wait and when to go. The closest living example of this I can highlight would be Hobo symbols that survive and are still in use today in the US. Different symbols would mean different things to different groups and insider knowledge for understanding thieves' cant symbols would be a must.

[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.