Does an object rotate around its COM when no torque is given?
For example if I were to throw a baseball bat in the air(I will hold it by the end of its shaft while I am throwing it and swing it upwards in order to give it angular momentum, and after I release it, the only force that acts on the bat is gravity),
Would it immediately start to rotate around its COM the moment I release it?
Yes, it would, since you applied a torque to it initially. You can think about it in terms of forces and linear momentum. If you accelerate an object to a certain velocity and then stop accelerating it, it will have a new momentum than from right before you started accelerating it in the first place. If you apply a torque to the baseball bat when throwing it, you are actively changing its angular momentum. When you release it, it retains its angular momentum just as an object retains its linear momentum after being accelerated. If the object has an angular momentum from a torque applied to it, it will rotate around its center of mass.
When the total torque about the center of mass is zero, then there is no angular acceleration. If the object is already rotating, it remains rotating. If it is not rotating, it doesn't start.
Net torque equaling zero is not enough information to tell you if an object is rotating or not.
