Do Moment exist for objects in no-gravity enviroment? Imagine a stick with 2x lenght. with mass of m. 
And COG is half way through.  This stick in space. No gravity and friction exist
I pushed the stick from one end point. Do stick start rotating? Or it just go forward without rotation because there is no pivot?  
 A: No, Newton's laws apply just the same with or without gravity as long as you find yourself a comfy inertial frame to do your experiments. So as long as you're observing from an inertial frame, if you apply a force on a stick from one end, you'll observe it rotate because the force, not passing through the center of mass, would provide a torque about an axis passing through the center of mass, causing it to rotate about the center of mass. 
In fact, uniform gravity plays no role in rotating an object because such gravity would accelerate all the parts of the body just the same, which means no contribution to rotational motion.
Response to Comments
That's a compelling intuition I have to admit but it's simply not supported by physical principles of motion. If I roughly talk in your language then even if you don't have the force of gravity there is a kind of "imbalance". The imbalance is that you're applying a force that is not uniformly distributed throughout the stick, in particular, it's not distributed symmetrically about the center of mass. An example that should help you change your intuition: put a stick horizontally on a table. The force of gravity is cancelled by the reaction force from table so gravity plays absolutely no role here. Apply a horizontal force on one of the ends of the stick and it'd still rotate if the table is smooth enough. 
