A force is applied at different heights on a sphere kept on horizontal smooth surface. What axis it would rotate about in each case and how? The acceleration of the centre of mass will be F/m, no matter what the height of point of application of force. How would the rotation of the sphere be, what is the  axis of rotation?
Also, if it does rotate, then the work done on covering a certain distance, say, x would be Fx, and in some cases work done would increase the translational kinetic energy, but in some cases it would increase both translational and rotational kinetic energy. (I think there's something wrong with this assumption. I just cannot understand it). Then, same work done if gets divided in two parts, would result in a decrease in translational velocity. Then the earlier conclusion that all spheres have same acceleration and velocity of COM would become wrong. What would be the axis of rotation, if the body does rotate, does the rotation occur in a way that the velocity of COM after adding both translational plus the tangential velocity due to it, become equal in every sphere?
 A: The net force divided by the total mass would cause the acceleration of COM by dividing the net force by sphere mass, and from COM acceleration, its velocity can be calculated through integration. So the center of mass velocity, and hence translational kinetic energy, should not be affected if the forces caused rotation or not.
On the other side, the net torque around the center of mass would cause the rotational acceleration around COM by dividing net torque by sphere moment of inertia,  and from rotational acceleration, the sphere angular velocity can be calculated through integration. So the rotational velocity, and hence rotational kinetic energy, should not be affected if the forces caused translation or not.
So the translational speed would not be decreased if the net force caused rotation, instead, a greater amount of work is done by various forces such that a part of this work is converted to the translational kinetic energy and the other part is converted to rotational kinetic energy.
To summarise:
work done by forces, in case of both translation and rotation > work done by forces, in case of pure translation
Total work= Total Kinetic energy = work done by forces torques around COM + work done by forces throughout the total COM translational trip = Translational kinetic energy + rotational kinetic energy
