There are plenty of similar questions here, but I will nevertheless add another one because I believe this is the ultimate source of confusion (and will probably provide a one-stop solution for future questions of similar nature) -- what is so special about the center of mass when it comes to the rotation?
You have a rod in space. You provide an impulse to one of it's ends. It's COM starts translating at a uniform speed, and the rod simultaneously starts rotating (to conserve angular momentum) about its COM (okay, about an axis through COM); why just the COM? Why not any other point? (Same goes for 2 masses connected by a massless rod, an impulse on one of them causes rotation about COM)
Possible answers -- The COM is stationary (if not, choose a frame moving with the same velocity as the COM), so only it can provide an axis of rotation in an inertial frame. Another possible answer (not necessarily correct) -- The rod's motion can be thought of its translation plus rotation about ANY point on the rod; choosing the COM makes the analysis easy. Are any of these explanations correct? Is there a better one?
You have a rod (again, free), and now you continuously apply a constant force of equal magnitude and direction, on both ends. Intuitively, you wouldn't expect the rod to rotate. You say 'at all times, the torque about the COM is zero, so the body will not rotate'. But at any instant, about ANY other point, there is a non-zero torque. Why are we only considering torque about the COM? Why not any other point on the rod?
Possible answer (not necessarily correct) -- For every point on the rod where the torque is nonzero, there is a symmetrically located point on the other side of the COM with an equal and opposite torque (should you calculate it), so by symmetry we should expect no net rotation.
The idea is, it would be great if someone could clear these common doubts by either elaborating/pointing out the mistakes in the 'possible answers', and providing the right explanations. I believe I have tried to condense all common doubts of similar nature in the question, at the expense of poor readability.