For Hess' law type questions, when do you take the inverse of the constant and when do you switch the sign?
Equilibrium constant

When a reaction is at equilibrium, the species will have concentrations called the equilibrium concentrations. They do not depend on whether the reaction is written in one direction or in the reverse direction, and whether all the stoichiometric coefficients are multiplied by the same value. However, your equilibrium expression will be different (switching concentrations from numerator to denominator if reversing the chemical equation, or changing exponents when changing stoichiometric coefficients). As a result, the value of the equilibrium constant will change while the equilibrium concentrations themselves will remain the same.

Enthalpy of reaction

For a given reaction, the enthalpy change associated with making - say - 1 mol of species A is a given. If you use 1 mol of species A in the reaction instead, the enthalpy change will be of the same magnitude but opposite sign. If I make a different amount of species A, the enthalpy change will change in a manner proportional to that amount. In a similar manner, if I change the chemical equation for $\ce{A -&gt; B}$ to $\ce{2A -&gt; 2B}$, the molar enthalpy of reaction will double as well. The same goes for Gibbs energy or for entropy of reaction.

Cell potential

For a given reaction with certain concentrations of species in the two half cells of a voltaic cell, I will measure a certain cell potential. If I switch the role of reactants and products (in the equation), I will still measure the same potential (the direction of the actual reaction does not change). If I multiply all coefficients in the chemical equation by the same number, it still correctly describes the reaction, but this will not change the potential.

Mathematical relationships

$$\Delta G^\circ = - R T \ln(K)$$

The way logarithms work, if you multiply $\Delta G^\circ$ by a number, you have to raise K to the power of that number. If you add Gibbs energies, you have to multiply equilibrium constants.

$$\Delta G^\circ = - z F E^\circ $$

If you multiply $\Delta G^\circ$ by a number, z (the number of electrons transferred) will also increase by that number. Therefore, $E^\circ $ does not change.

Multiplying all coefficient in a chemical equation by (-1) changes the role of reactants and products. Thus, once you know the rules for changing coefficients by a common factor, you don't need to memorize another rule for reversing the equation.