1. I don't agree. I think the roots of language are very firmly there in "animals". This may appear trivially true to you, but so often do I hear the phrase "What distinguishes us from animals is this or that or some other." that I always feel I must object to what amounts to rank anthropocentrism.* (I realize you haven't exactly said this.)
In this case the difference is so marked, that it's perfectly justified. Sorry, animal friends, but that's the reality.
2. What animals do, in language terms, appears very simple to humans. I think it's probably very complex.
No, it is objectively entirely primitive to human communication (a.k.a. language), that's why we reserve the term language for the complex communication systems humans have developed.
If I were to use an analogy, what animals do in comparison to what humans do is as pushing a ball forward and backwards with your snout is to the game of football, times one hundred. Animal communication usually consists of sending pretty straightforward information - "look, a threat!" or "here is food" or "back off, I am angry!" or "follow me", things like that. We have
some evidence that
some species of animals may be capable of more - maybe communicating additional information concerning the nature and magnitude of the threat, quality of the food, more nuanced emotional states, perhaps even things like numbers, directions, etc. And by all means, that's probably how human language started, millions of years ago. The point is, we have since evolved something that absolutely outperforms the earlier forms of communication. The amount of information we can communicate is huge compared to what animals can communicate using their mental faculties and forms of communication.
3. It is indeed stunning. How do you explain it?
Specialization and natural selection. Humans started out with basic forms of communication, but gradually improved on it because it was their main 'competitive advantage' in comparison to other animal species. The increasing performance of human brain then allowed for more complex forms of communication, which in turn created further pressure on the brain to evolve. We're a result of nature's runaway evolutionary experiment, I guess.
BTW, we now have evidence that the Neanderthals, the closest species to Homo Sapiens (or a sub-species of it, depending on one's point of view) had language. So, it's not a uniquely Homo Sapiens trait and it has roots in earlier species of humans.
I really think you are wrong. One article I was reading proposed a way to measure complexity of language by calculating how many sequential sounds could you use to add accuracy to a prediction of the next sound. This can be done with no knowledge of the language itself. Off the top of my head, primitive indigenous languages were around 5, English was 8, dolphin sounds were 9 and humpback whale song was 11.
Which only proves that it doesn't prove anything. Human languages can produce an infinite amount of utterances (the things you say) using a limited set of building blocks.