There is perhaps Pauling's principle of electroneutrality "each atom in a stable substance has a charge close to zero" - maybe that's the principle referred to in your book? "Close to" is an important part, though, and this is not some law of physics, just an observation of how things tend to be, not how they must be.
Consider also a capacitor - that's how the membrane behaves. A capacitor features charges that are separated by some insulating medium.
Unfortunately, some textbooks do a really terrible job of explaining how membrane potentials work. I recommend starting from imagining no membrane potential at all, just ions of different concentrations, and then think about how those ions would move across the membrane according to those concentration gradients, ignoring charge entirely.
If, say, K+ is high inside a cell and low outside a cell, K+ is going to tend to diffuse out of the cell more often than it diffuses in. No charges needed to understand that so far, just concentration gradients. But, as a consequence of K+ ions tending to move out of the cell, you're creating an electric current: the K+ ions have a positive charge, so if a few more leave the cell than enter, the inside is going to be a little more negative than it was. And since it's a little more negative, there will be a little less tendency for K+ ions that have positive charge to leave that slightly negative space, even if the concentration gradient would make you expect K+ ions to keep moving.
How much more negative? And how many ions will move until it's "negative enough" that the electrical gradient holding K+ ions in is equivalent in strength to the concentration gradient letting K+ ions leak out? I posted some of the math for the number of ions here - as you mentioned, it's very small. Electricity is powerful stuff, and that's why Pauling has that principle that things tend to be roughly equal. For a membrane potential, things are equal enough that if you were just writing down the concentrations it would fit into the measurement error. For the "how much more negative?" question you can use the Nernst equation.
When multiple ions are involved you'll have to weight each of their contributions to calculate the overall membrane potential. Only ions that have some permeability across the membrane contribute to the membrane potential. You can both calculate the membrane potential and also see how ions that don't have any permeability don't contribute from the Goldman equation. You can also see it from the story above: the voltage across the membrane caused by potassium is only happening because potassium is moving.
When someone talks about "injecting Na+", they don't have a vat of sodium ions, rather they're using this for shorthand of "injecting Na+ and an unimportant anion", that is, something that's not important for the membrane potential.
Be careful about trying to add up charges on each side of the membrane to calculate a membrane potential: you want to think about concentration gradients, instead, and how ions move according to those gradients and what consequences that has for the distribution of electrical charge across the membrane. You also shouldn't think about the charge that moves across the membrane as being distributed throughout the cell: it's literally at the membrane. Remember, the membrane is a capacitor: the extra charge exists right along the membrane, and it doesn't just sit there without having an effect on anything, a bit of negative charge inside the membrane is going to pull any positive charges anywhere in the membrane a little closer to the inside, and push any negative charges a little further away.
