A physics engine updates position from velocity and velocity from acceleration, each step multiplying by the timestep. Why does halving the timestep improve accuracy, and why does it not eliminate the error?

C. Because the update assumes velocity is constant across the step.

Each update treats velocity as constant over the step, so it misses the change happening within it, and a shorter step improves the assumption while doubling the number of steps, so the improvement is a net effect. It never reaches zero, and below some timestep accumulated round-off grows faster than truncation shrinks. Which is why engines use Verlet or Runge–Kutta rather than plain Euler.