I think this isn't too bad, and your guess in Q2 is justified. I doubt it depends on the detailed formulae.
Suppose we concentrate on g - f, which is constrained to have integral 0? And by a change of origin to have a graph that is non-positive on the negative real axis, and non-negative on the positive real axis. The main issue is that multiplying it by the independent variable (x in the current notation) you will get a non-negative function, so a non-negative integral. I believe this is what you want.