Assumptions for Nernst Equation
Assumption (1) is not really even true. The Nernst equation is always valid to find the reversal potential for a single ion: that is its purpose. To determine the equilibrium potential for a number of ions, you use the Goldman/GHK equation. It happens to be that the Goldman equation reduces to the Nernst in case of single-ion permeability, which of course never happens for real but is a reasonable approximation in some cases.

As far as your additional assumption, no, that assumption is not required. The Nernst equation doesn't tell you which directions ions flow, it tells you what the reversal potential is for an ion. If you add a voltage from some other source (like your example of adding a bunch of mysterious unaccounted for positive charges outside the membrane) that doesn't change the reversal potential, it changes the voltage. If you put a large positive charge on the left side somewhat like your situation (B), potassium would flow up its concentration gradient towards the Nernst potential for potassium.

Note also that your example is utterly ridiculous: if you want to know what happens when you have 10000000 M Na+ ions together on one side of a membrane you need to consult a physicist, not a biologist. I don't know where you put all the other negative charges, but you're probably about to have a) A very expensive electric bill, and b) a very big hole in your lab.

In real biology, the number of charges on each side of the membrane is similar enough that you can call it equal (see for example Why is it possible to calculate the equilibrium potential of an ion using the Nernst equation from empirical measurements in the cell at rest?). In any case where someone describes "adding sodium" to one side of the membrane, they don't mean just sodium, they mean sodium as a salt with some negative ion, it's just that the identity of that negative ion doesn't matter if it isn't membrane permeable (relative to the other ions involved).