In general in play they were ignored or just treated as an abstract language with no further comment.
As to where they came from, here's an answer from Gary Gygax on Dragonsfoot!
As D&D was being quantified and qualified by the publication of the supplemental rules booklets. I decided that Thieves' cant should not be the only secret language. Thus alignment languages come into play, the rational [sic] being they were akin to Hebrew for Jewish and Latin for Roman Catholic persons.
I have since regretted the addition, as the non-cleric user would have only a limited vocabulary, and little cound [sic] be conveyed or understoon [sic] by the use of an alignment language between non-clerical users.
If the DMs would have restricted the use of alignment languages--done mainly because I insisted on that as I should have--then the concept is vaible [sic]. In my view the secret societies of alignment would be pantheonic, known to the clerics of that belief system and special orders of laity only. The ordinary faithful would know only a few words, more or less for recognition.
In other words, it was supposed to be more like religious languages, but wasn't really well thought through. It disappeared in Second Edition and was not missed.
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
The psionic power mindlink does exactly what you want: It's low level and works on any creature that has an Intelligence of 3 or higher. (As long as you spend the points to make it work on an unwilling target.)
And most importantly:
As a backup option, as long as you can touch the ice toad, you can cast tongues on it. Tongues doesn't require a willing creature, so as long as the ice toad fails its will save this should allow it to speak to and understand you.
This won't allow it to speak to other ice toads:
So if you want it to be able to speak to other ice toads, you'll have to cast the spell on them too.
There is one other fly in this ointment:
This is potentially problematic, because if ice toads don't have a voice (which may or may not be the case), they will be able to understand you but not speak loud enough for you to hear. This will enable any communication method that relies on the creature understanding you, and might allow communication methods that rely on the creature having a language. (Depending on whether understanding every language counts as "having a language"; I can't find a rules definition of what that requirement actually entails.)
Alternatively, with one-way communication established, you may be able to talk it into accepting a telepathic bond or some other method that relies on having a willing target.