There was no work done by friction in your example. Work is force by displacement. Friction is between men's feet and the board they contact. During this contact, each foot does not move relative to the board. Therefore the displacement is zero and so is work by friction. Actual work here is done by men's muscles by converting the chemical energy of food to mechanical energy.

The kinetic energy is defined the way you stated. The sum of kinetic energies of all bodies is non-zero, but the kinetic energy of the center of mass is zero. There is no contradiction here.

There was no work done by friction in your example. Work is force by displacement. Friction is between man's feet and the board they contact. During this contact, each foot does not move relative to the board. Therefore the displacement is zero and so is work by friction. Actual work here is done by man's muscles by converting the chemical energy of food to mechanical energy.

The kinetic energy is defined the way you stated. The sum of kinetic energies of all bodies is non-zero, but the kinetic energy of the center of mass is zero. There is no contradiction here.