A fascinating study - it seems the more we learn of the Universe the more beautiful it is, and the more questions it poses.
A point I don't understand is that phrase, "oxygen burns". I know both that oxygen itself is not flammable, and that stars work by nuclear-fusion, so I don't know if what they mean here is either the oxygen is consumed in oxidising other elements (e.g. sweeping up remnant hydrogen atoms to form water) or its atoms fuse into the next element along the Periodic Table.
I have an idea I have heard astronomers use the word "burning" rather loosely to mean nuclear fusing.
A lot of the gas and dust floating about in galaxies is elements, but there also simple compounds, so the atoms are combining somewhere. I don't know if this has been established but I would have guessed the outer layers or atmosphere of stars, where the heat and pressure are insufficient to fuse atoms, allowing them to combine into molecules.
Oxygen will hook up with almost anything going - so water, carbon-dioxide, silica and other rock-forming minerals, and so on.
It's fusion. An "oxygen-rich" stellar core is mostly carbon, but retains sufficient energy and mass to continue to fuse and transmute heavy nuclei into still heavier elements - mainly iron, hence the "Fe core collapse" in the next stage on the diagram.
I've an idea I read that going further beyond Iron is quite a major step, and a lot of more modest stars end at that point.
I have just realised the significance of something linked to this, but which I looked up when answering one of Rosie's questions the other day.
All that nuclear fusion, and any chemical reactions among the generated elements, is making the raw materials for planets and the things on them. Among those elements is Uranium, whose main isotope (I found for her question) is not ever so strongly radioactive but has a staggeringly long half-life. If it were not so, I wonder what Earth would be like now? Its Mantle is molten but extremely viscous rock, kept hot by the decay of uranium and other radioactive elements. If all the Uranium from the planet's accretion had become lead by the time the Earth was even half its present age, would the Mantle have cooled so much that Continental Drift and volcanoes could not exist?
Great questions! :D These would be worth asking the physicists. They'd have the maths to calculate it. And they could work with geologists to get core samples for further proof.
Element 99 