Why DC voltages in series do sum (or subtract, according to polarity)?

It's not really a difference in charge, it's a difference in the energy of the charge. An increase in voltage between two points means that the charge (electrons, usually) at the point of higher voltage have more energy than those at a lower voltage. As an electron moves through a battery from the - terminal to the + terminal, a chemical reaction in the battery adds energy to the electrons. For some batteries, you can make electrons flow into the + terminal and exit at the - terminal. In that case, the electrons lose energy when they cause a chemical reaction that recharges the battery. Note that the electrons will try to move to a place where they have lower energy, so current always flows through a resistor from the terminal with high voltage toward the terminal with low voltage and the energy of the electrons is always expended as heat.

You ask "what is going on between B and C?". Nothing. They are electrically the same point in the circuit, and they have the same voltage by definition.

Also, just to be pedantic, there are no electrons flowing in the circuit you drew because there does not exist a complete path for their flow. Although no current is flowing, the charge at different points has different levels of energy.

For this case you can visualize Voltage not as the amount of electrons flowing but the pressure behind them. In this analogy the electrons are like air, pumped around by the batteries. Note that in this case the batteries are constant-pressure-adding pumps, not constant-flow pumps.

In your case battery 1 pumps electrons from D to C, adding a pressure of 9V. If you wire a lamp from C to D, the battery will force electrons through the lamp with a pressure of 9V.

battery 2 will pump electrons from B to A, adding a pressure (at B) of 9V (relative to A). When we regards things relative to D, the electrons at C and D already had a pressure of 9V, so at A they will have a pressure of 18V (relative to D).

Note that we don't care what the absolute pressure at D (or any other point) is. Electrical Voltage is always relative to some assumed 0 point.

If you wire a lamp between A and D electrons will circle round the loop, from A through the lamp to D, pumped to C, through the wire to B, pumped to A, and back into the lamp.

I talked about electrons here to simplify things, the more accurate term is charge. Charge can also be conveyed by ions, and an electron that flows into a wire might is not likely to be the electron that goes out at the other end.