"learnings" to talk about things that were learned but that were not taught

Learning as a noun is very unnatural. Using it would instantly mark you as someone who speaks English as a second language. The appropriate noun to use is lesson. I don't think there is any logical reason why it is unnatural, because, as is so often the case, language is unpredictable.

You say that lesson or lessons implies intentionality, but this is not the case. Neither does the verb to learn imply an intentional teacher. This is asymmetrical: both to teach and teachings are intentional, but it is equally valid to say that something was learnt in intentional and non-intentional contexts.

The ability of fish to breath in water would not be considered something that is learnt - it is simply innate. Other animals certainly do learn, and can be taught. As with humans, it is appropriate to say that animals can learn skills even when not being intentionally taught.

"there is nothing wrong with teachings so why with learnings?"

To which I should respond, it might be nice if English had such an expression, but it doesn't.

The good news: "Lessons" can cover a lot more than formal sessions of teaching: you have "life-lessons". "Never do such-and-such at poker", and after losing your shirt you have learned a important lesson. So I assure you that you can use "lessons" for your 2) and 3) without fear of committing yourself to their being taught by some guy with chalk-dust on his jacket and leather patches on his elbows (or with horizontal green ears...).

I am unhappy with the idea of the fish "learning", but this is another issue. Come back to that if you want.