A person falls when a carpet is pulled from below their feet. But for an object to fall, a couple should act on it. One force of the couple is provided by the friction between the person's feet and the carpet. But what is the other force?
There is no required couple in this situation, and if there were a couple moment, then person would not fall down.
A moment couple is when the net force on the body is 0. This leads to net rotation without translation.
When a moment is applied to a body, there is no requirement for a coupled moment if the body is also free to translate.
In this case, the body is free to translate; and the rotation causes the centre of mass to go lower as the rotation occurs. There is both translation and rotation.
So really, I think this boils down to a problem with the statement:
"But for an object to fall, a couple should act on it. "
Which is basically the opposite of what really happens. In reality, if there is only a couple acting on the object, it will not fall (or change it's motion) at all. The net force is 0 so the change in momentum is 0.
Due to force of friction the feet move in forward direction but due to inertia the upper part of the body remains at rest so line of action of weight of the person passes through behind the feet. This constitute a torque by virtue of which the person falls behind.
The other force is the pseudo force. When you pull the carpet which was initially stationary, you actually accelerate it. So the person suddenly finds himself in an accelerating frame, hence a pseudo force starts acting on the person in the direction opposite to the acceleration. The magnitude of this pseudo force is $ma$ where $m$ is the mass of the person and $a$ is the acceleration of the carpet. Due to this force, the person tries to move away from the carpet. However, the friction between person and carpet tries to oppose this motion and starts acting in the direction opposite to the pseudo force. These forces form a couple and the person falls.
Extra note: the pseudo force acting on the person acts on his center of mass. Hence a person with a lower center of mass is less likely to fall than a person with a higher center of mass for a given acceleration of carpet.
It depends on the point you choose as reference to torques. If you take the CoM, there is only friction, but a force can be enough to provoke torque because there are internal forces of reaction inside your body. Just like the way you turn a wheel with one only hand.
But you can take more references, such as your knees. Then the weight would indeed cause torque but the magnitudes and distances are different. You will finally arrive to the same conclusions but some references are easier than others.