Ultimately this is more of an overly long comment, as I think a more satisfying and complete answer would properly explain things in a more concrete fashion—more of a "it has to do this because..." answer than my "it can do this because..." one.
The short of the answer to the first question is that helium fusion needs ~25 times the temperature that hydrogen fusion does. The proton-proton chain initates around $4\times 10^6$ Kelvin, whereas helium fusion doesn't begin until around $10^8$ Kelvin. So when the main stage ends and the helium core contracts and the temperature rises, the "edge" of the core can have temperatures well in excess of the minimum hydrogen fusion temperature, and so a shell around it can have temperatures well beyond it. Fusion rates are (approximately) polynomial in temperature, with the degree depending on the reaction in question, so small increases in temperature can produce substantially more fusion. The gravitational force is strong enough to overwhelm the pressure from the induced fusion, and so will contract the surrounding shell to temperatures in excess of the minimum needed. This is basically what happens in the cores of main sequence massive stars (relative to, say, the Sun). Actually, our own Sun's energy is mostly from the proton-proton chain and has a core temperature of around $1.57\times 10^7$ Kelvin, nearly four times the minimum necessary. And still, the core needs to be nearly 10 times hotter than that to initiate Helium fusion.
For the second question, the short of the answer is that the core has undergone thermal expansion after the Helium flash, and so occupies the temperature ranges most conducive to a strong hydrogen fusion rate. The material outside the core is now at lower temperatures and pressure, so the fusion rate is reduced substantially. So the energy output comes principally from the core, and the helium fusion at near minimum temperatures releases less energy than the hydrogen shell at (well) beyond minimum temperatures did. Thus the star overall produces less energy and contracts.
The remaining questions are explained in similar fashion: one has to pay attention to the sensitivity of reaction rates to temperature, and what the temperatures in those shells actually are. The sensitivities are different for each reaction chain, and the temperatures can go well beyond the minimum necessary.