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.