It seems to me that the definition of "species" is not clearly defined. I've heard it said that the defining characteristic is place and time. So even though a spotted owl may be able to mate with other owls, it is a separate species because it has been in a different place for a long time.
As an illustration of new species generation, consider the Brazilian rainforest. It is said to have existed for 50,000 years and is a habitat to about the same number of species that would become extinct if the rainforest were to go away. So that would mean that one species evolved every year.
1) Like you point out, a new species couldn't evolve abruptly because it would have no mate. But donkeys can mate with a horse producing a mule (and some types of mules can breed) even though they are genetically different. Some animals can breed with others even though they have different numbers of chromosomes.
That would indicate that the process of evolution can be a very gradual and continuous one like Darwin said. Each animal could be a mate with its predecessor or successor but the first could not mate with the last. The problem is a lack of fossil records of continuous evolution. I think this is because the intervening links would be inefficient and have very low populations. We seldom find their fossils and when we do we just call it another species.
This is reasonable because if we have had 10 million generations for a million species that would make 10 trillion fossils. But we have uncovered only a tiny percentage of them so the very tenuous links of low population would be obliterated with time and not visible to us -- all we see are radically different fossils. It's like looking at a photograph of a tree. If the photograph is out of focus and underexposed we see just a group of dots instead branches.
2) At first I thought the concept of something that has no beginning was kind of silly. But then I realized that a circle doesn't have a beginning or end. So maybe what you are getting at is maybe world lines are circular. Maybe the last man alive in the universe invents the first cell of life. It could happen.
As for government schools teaching this stuff -- I don't believe in government schools, I believe in home schooling.
As for government schools teaching this stuff -- I don't believe in government schools, I believe in home schooling. On this we are in total agreement. Unfortunately we are in the minority and there are government schools and so long as there are, they have no business mucking around with origins when the numbskulls called teachers cannot even teach mathematics, english, geography, and history.
As for the rest of your post, you have made good points, which I would address one at a time, but there is nothing conclusive I could offer, only my opinion and reasons for it. I do not think we are far apart, but our conclusions are a little different.
I did not mean by, "no beginning," circular, but infinite. In spite of the famousness of the "big bang" theory, it is not nearly as conclusive as people believe, and many scientists and philosophers throughout history believe the universe may be both infinite and eternal (although "indefinite" would probably be the better word since both infinite and eternal imply some kind of metric or measurement). Even if one believes in the "big bang" that does not mean nothing came before, or that where the "big bang" takes place, there is nothing anywhere else (whatever that would mean in that context). A beginning is just one view among many possible views (and your circular idea is actually entertained by some phsicists), and as silly as "no beginning" seems, many of the scientific "facts" now commonly accepted seemed pretty silly when first proposed. None of these conjectures may turn out to be correct, but so long as they are possibilities, the certainty of any one hypothesis of origins is not established.
Hank