What assumptions underlie Hess' law and and its analogues?
The typically unstated assumption in applying Hess's law is that you are going from an initial state of pure reactants (say, in separate containers) to a final  state of pure products (also in separate containers).  In the case of ideal gas reactions, the enthalpy of mixing is zero and enthalpy is independent of pressure.  So, for ideal gas situations, Hess's law can be extended and applied directly to starting and ending states in which the reactants and products are mixed (and even simultaneously present) and the partial pressures are all different from the standard pressure of 1 atm.

Of course, since U, S, and G are functions of state, the strict version of the law, applying only to pure reactants and products, still holds.  However, for ideal gas mixtures, partial molar entropy and partial molar free energy are functions of mole fraction, so this needs to be taken into account for mixtures.