Decreasing a digraph's edge-weights while keeping net weights of edges at each vertex constant
The second objective is just a simple circulation problem http://en.wikipedia.org/wiki/Circulation_problem
as the maximum circulation gives the highest reduction in weight. This problem is basically equivalent to max flow for which efficient algorithms are well known. 

Not so sure about trading of the first and second objective. You might achieve something like this by modelling it as a min concave-cost network flow problem with capacities, e.g. by some cost function that is nearly flat at arc saturation, such that it favours saturating arcs once they are close, but this is NP-hard:
http://link.springer.com/article/10.1007/BF02283688

Maybe there is a more efficient way depending on what you are trying to achieve,  but the constraint that net edge weights remain the same indicate that you are looking for some network flow type algorithm. 

Do you have an application for this in mind?