>The real mystery, for me, is: Why was the initial state of our universe so completely non-chaotic? Why did the universe not start out in a state that is essentially equal to the heat-death [1]?
taken as a closed system, our Universe's initial state was a heat-death and it would stay here, nothing would happen. Fortunately or unfortunately our Universe is embedded into previous very old and cool Universe, and this gradient is what drives the expansion of our Universe and thus provides for all the things we see happening to happen, including life (which is just entropy maximizing process in some conditions).
100-200+B year later our Universe will too be very cool and expanded very "thin". The matter will be collected into several hyper-super-black holes with Schwarzschild radii of tens or even hundreds o billions of light years - some probably bigger than our visible Universe (which has 10B light years Schwartzild radius). The matter inside a regular black hole is neutrons being driven into higher and higher states as result of mass being added into the black hole. At some point the neutrons get to split into gluons/quarks and those would continue to be driven into more energetic states as more mass is added, and may be split as a result into something else too. At the same time the space around becomes "thinner" and cooler. At some point, when even more mass is added, and thus this "soup" becomes even more energetic, beyond currently known physics unfortunately, something happens to the gravitational force what has been keeping it together and it explodes and start expanding (as there is extremely high gradient between the "soup" and the space of the old Universe - thus that will be a birth of new Universe inside our old Universe, inside the parent Universe of our Universe ... ad infinitum ... with non-zero gradient, though smaller and smaller as parent-child Universe pair is getting older...