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.
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.
Best Answer
Yes
When you 'learn to do' something that doesn't mean that you are forced to do it. The ability makes it clear that you are merging your ki to translate the message, and just because you know how to do that doesn't mean you are forced to do that.
The whole point of ki is control of one's self.
Generally beneficial abilities like this have absolute wording, most likely (IMHO) because the number of situations where it isn't beneficial is limited and the designers didn't think about them. Going purely RAW without putting any thought into the how or why isn't how the game is intended to be played.
There is no 2 1/2 "The DM rigidly translates every word on the page."