How to calculate membrane potentials?

I have recently been reading about membrane potentials in the cell and how they are derived. The textbook (Guyton and Hall Textbook of Medical Physiology, 13th Ed) gave this equation: $$EMF=\pm\frac{61}{z}\log\frac{C_i}{C_o}$$ Where $$i$$ refers to the interior of the cell, $$o$$ refers to the exterior of the cell, $$z$$ is the relative magnitude of the charge of the cation/anion, $$EMF$$ is the potential in the cell, taking the external potential as 0. Additionally, the sign is + for anions and - for cations.

I understand that this comes from the Nernst Equation: $$E=E^o-\frac{RT}{\nu_eF}\ln Q$$ Where $$\nu_e$$ is the number of electrons transferred per mole of reaction.

I understand that $$E^o=0$$ because at equilibrium, the concentrations of ions on both sides of the membrane are equal and I also understand where the magnitude comes from: $$61=\frac{1000RT}{\log eF}$$ Where $$\frac{1000}{\log e}$$ is used to change the base of the logarithm and change the units to mV and T is taken to be the body temperature of 37 C, converted to Kelvins.

However, what confuzzles me is

1. The sign convention: Why are cations - and anions +.
2. How do we derive the $$-\frac{RT}{\nu_eF}\ln Q$$ for transfer of ions, since unlike the Nernst Equation, which is derived for electrons, ions are being transferred instead?
3. How do we measure the membrane potential? According to the text, they put one electrode in the extracellular fluid and pierce another electrode (a typical Ag/AgCl electrode) into the cell, in contact with the intracellular fluid, as shown below. Since there is no transfer of electrons, how does this apparatus actually work?