The shell burning would be CNO cycle even in the subgiant phase for most stars, because remember, the solar core is already almost hot enough for CNO cycle fusion to dominate, so it wouldn't take much contraction and heating to get CNO to take over. But the basic story given above is correct-- as the core contracts, the shell gets hotter, fusion in the shell goes berserk, and heat is deposited in the envelope, expanding it. The expansion serves to take weight off the fusing shell, lowering the amount of mass in the shell and turning down the amount of fusion, thus maintaining equilibrium with what light can diffuse out through the shell. (Note that the luminosity ends up rising even as the shell gets less and less mass in it, because its temperature is going up, and the rate that light can diffuse out also goes up as the amount of mass in the shell drops, so it is always in equilibrium as the helium ash builds up in the core).
On the matter of the transition from sub-giant to giant, I would say that's the transition from an ideal-gas core to a degenerate core. This comes with a very significant shrinking of the core, such that at first the core and the fusing shell above it constitute a significant contribution to the radius of the star, so a significant part of the star is transporting heat by radiative diffusion. However, as the core and shell shrink and the envelope expands, most of the star is transporting heat by convection. The difference between a primarily radiatively diffusing star, and a primarily convecting star, is that the former evolves at nearly constant luminosity (i.e., horizontal), and the latter evolves at nearly constant surface temperature (i.e., vertical). These phases are also seen in pre-main-sequence evolution, in the reverse order. The fully convective interior is called the "Hayashi track," and the more radiatively diffusing interior is called the "Henyey track."
A small correction on the Schonberg-Chandrasekhar limit-- intermediate mass stars (say 2-10 solar masses) are the ones that encounter this limit, they are not the ones that don't encounter it. It happens whenever the inert (i.e., non-fusing) core is still an ideal gas, so has not yet become degenerate, and also acquires more than 10% of the star's mass. For the Sun, the post-main-sequence helium core is degenerate by the time it has 10% of the mass, so there's not the rapid core contraction that typifies this limit. That basically means the Sun stays a subgiant for a little while as its core contracts only slowly, allowing the star to remain mostly radiatively diffusing, before puffing out a huge fully convective envelope. But higher mass stars have convective cores, so the hydrogen runs out over the whole core at once, allowing them to still be ideal gases when they have more than 10% of the star's mass. That causes the rapid core contraction, so they essentially have their subgiant phase "skipped over", creating something known as the "Hertsprung gap" in the H-R diagram.
