Why are non-spontaneous reactions driven in the presence of energy? Here is my understanding. If you have an endergonic reaction (m) with $\Delta G_m=-m$ and you couple it to an exergonic reaction (n) with $\Delta G_n=n$, where $n&gt;m$, then the endergonic reaction can proceed as the overall $\Delta G$ for the system is negative. I understand why having the endergonic reaction occurring is not a violation of the second law of thermodynamics and why it is made possible in that sense.

I was recently studying electrolysis and they talked about hooking up an electrochemical cell to a power supply and using electrical energy to drive a nonspontaneous redox reaction. If you include the power supply in the system, then the $\Delta G$ is negative, as the power supply gives a lot of energy.

This is my confusion: if the redox reaction is nonspontaneous (let’s call it R1), then it must be spontaneous if the opposite direction (let’s call it R2). However, in the presence of electrical energy… R1 is made spontaneous? Therefore it can occur? But what of R2? Shouldn’t that reaction be even more spontaneous, as it were? Shouldn’t the system try to minimize its energy by preferentially allowing R2 to proceed, even in the presence of an energy source? Thus, why would R1 occur? Why does shoving energy into a system cause a nonspotaneous reaction to occur when it would be more favorable to continue having it occur in the opposite direction?

I would also appreciate it if anyone could extend this to general energy coupling, say with hydrolyzing ATP and a nonspontaneous biological reaction. Why is the original spontaneous reaction not made more spontaneous?
You are incorrect in your assumption that the addition of a power source makes both the forward and the reverse redox reactions more spontaneous. An electrochemical cell operates on the same principle as a battery. On the anode of the cell, you will find a compound that is being oxidized, or losing electrons. These electrons then flow through some sort of a wire to the cathode, where the electrons are being used to reduce another compound. The electricity will flow in the "spontaneous" direction, in other words, from the compound that prefers to be oxidized to the compound that prefers to be reduced. The "voltage" across an electrochemical cell is determined by adding up how badly the reducing agent wants to get oxidized ($E°_{ox} = -E°_{red}$) and how badly the oxidizing agent want to be reduced ($E°_{red}$), and is a measure of how much the reaction wants to occur. 

Now, lets try adding a battery in between the anode and the cathode. If the battery is added such that its anode points towards the cell's cathode, the battery will add to the voltage of the cell, causing the effective voltage for the reaction to increase. On the other hand, if the battery is placed such that its anode points towards the cell's anode, it will be creating a voltage in the other direction. When this happens, it becomes a question of whichever voltage is stronger. If the battery has a stronger voltage than the cell, electrons will flow in the opposite direction. This is what occurs in electrolysis, causing the originally non spontaneous process to become spontaneous. This results in charging the electrochemical cell. On the other hand, if the battery has a smaller voltage than the cell, the electrons will flow in the original direction - charging the battery as they pass through it.

While a power source can be used to either make a spontaneous process more spontaneous or to reverse the flow of electrons, it cannot do both at the same time.