You're asking for more specific rules about how Tongue of the Sun and Moon works, but you've already quoted all of the relevant text. There are no more relevant rules.
This is pretty much entirely in the realm of "Ask your DM". 5th Edition is a lot less precisely specced than 4th, or even 3rd, and this kind of question about what happens in situations where something isn't called out directly in the rules is supposed to be handled by the DM. There are no mechanics for how "touching the ki of other minds" works on a spiritual level, so this is well within the bounds of DM rulings.
That said, in one of my games, I'd allow you to do basically whatever you wanted to do with that ability. I feel that class features should always be benefits, not hindrances, and that creative play should be rewarded. Want to talk to some people, but not others? Great! That sounds like an interesting and non-linear way of using that ability. Want to purposely not understand someone speaking Demonic, since they may be cursing you? That'd work too.
For your third question: I'd run it that you don't have to know that someone is specifically listening, if you don't want to. Like, you could say something to an apparently empty stadium filled with invisible people with the intent that everyone who can hear you can understand you, and everyone would understand you, even if you don't really know that they're there.
For your fourth question: You're not really affecting people with anything, per se. Since there's no mention of a range, or any kind of targeting constraints, I'd run it that you could speak and understand anyone, anywhere. If someone broadcast a message on a TV in a different language, I'd let you understand it.
[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".
- 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.
- No. A recorded audio message suffers from the same limitation as a written message: it is mindless.
- 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.
Best Answer
The PHB explains it on page 123: