How does an aircraft's weight affect the V-n diagram?
Well, answering your first question, the airplane weight does not really affect the V-n diagram because this diagram is developed knowing the structural forces that the airplane can take without being damaged. 

Answering your question about the relation between the maneuverability and the airplane weight, the relation is just a "simple" physics problem. You just need to know the airplane characteristics like the center of gravity, the different aerodynamic factors and the effect of deflecting the control surfaces. When you solve the six equations with six variables of the differential system you can find how any characteristic of the airplane affects its maneuverability.

 

Finally, as you say, when the weight is low, the needed lift is less but this does not affect the V-n diagram because n stays more or less constant because:
 $n=\frac{L}{W}$. This diagram just shows us which are the relation between the aerodynamic forces and the weight. We all know that the weight is one of the problems for an airplane but it also helps to alleviate the bending moment on the wing root, that is why the last fuel tanks that are emptied on an aircraft are the ones that are further away from the fuselage.