Points to Ponder

To figure out how the centre of mass of a system moves, you don't need to know anything about the internal forces between the particles. Only the external forces acting on the system matter.

A powerful trick in mechanics is to split the motion of a system into two parts: the motion of the centre of mass (the system's translation) and the motion of the particles relative to the centre of mass. For kinetic energy, this split is exact: $K = K' + \frac{1}{2} M V^2$, where $K'$ is the kinetic energy of the particles as seen from the centre-of-mass frame.

Newton's Second Law for an extended body (or any system of particles) doesn't come from thin air. It rests on Newton's Second Law applied to each individual particle, plus Newton's Third Law for the forces between those particles.

To prove that the total external torque equals the rate of change of total angular momentum ($\boldsymbol{\tau}_{\text{ext}} = d\mathbf{L}/dt$), you need more than just Newton's Second Law for particles. You also need Newton's Third Law, with the extra condition that the forces between any two particles act along the line joining them (central forces).

Having zero total external force and having zero total external torque are two completely independent conditions. You can have one without the other. A couple is the classic example: the net force is zero, but the net torque is definitely not zero. …