Confused with 'discharge potential',preferential discharge theory
I can answer your first two questions.


  I am confused by 'require', do ions upon discharging gain energy or lose energy?


Ions lose free energy during discharging, as you are referring to the spontaneous process of transferring charge at the electrodes with conversion of some of the ions into neutral species (in general you are speaking about the process described in the next paragraph).

For a process to proceed spontaneously, the free energy change for the system (combined half-cells, say) has to be negative. The Nernst equation tells us that the free energy difference is given by $$\Delta G = -nFE$$ where E is the total cell potential. This value must be negative for the cell to discharge, which means E is required to be positive. This is what is meant by "require" (language is used in funny ways, ions don't really require anything). You have more choices than just discharging the cell: you can charge it (creating more of some of the ions). Unless the potential is such to cause discharge you may actually produce more of the ions that are disfavored under spontaneous (discharging) conditions.


  are ions in general not less stable than neutral counterparts?


You should not attempt to generalize, as this depends on the context (details of the reaction in question). An ion may be the most stable state of an atom in one context (an oxidizing environment, say) whereas it will be unstable in another (a corresponding reducing environment). Moreover, you can apply a charging potential and thereby invert the sense of stability, rendering one species more stable than the other.