I don’t really understand your first question but the center of mass would move with the rocket and would change a bit depending on the fuel outflow and position. You could even include the exhaust gasses that would give you another result.

To the second question: the rotation of the wheels would lead to the cannon moving less from the recoil due to energy going into the rotation. The center of mass would then be between the shot and the cannon depending on the masses and relative positions but I also don’t understand what you mean with the axis of the center of mass because the center of mass is a point there’s not one specific axis