Are cells electrically neutral or charged to setup the membrane potential across them?
It isn't neurons that are charged, it is their membranes. From the distance of other cells, they "see" both the charges inside and outside the membrane, so there isn't any substantial repulsion on that scale.

The charge across a membrane comes mostly not from the Na+/K+ pump directly, but due to differential permeability of the membrane to different ions. Ions normally tend to flow down their concentration gradients, but that flow can be influenced by an electrical potential. From the perspective of just Na+ and K+, there is more K+ inside the cell and more Na+ outside, and more permeability of the membrane to K+ than Na+. Therefore, the inside of the cell will be negatively charged at equilibrium where the net flow of ion charges is equal. At a negative resting potential, the negativity of the inside of the cell equalizes the net flow of potassium out with the net flow of sodium in. However, the actual concentration changes involved are miniscule. From the perspective of ion counts, there are basically as many positive ions inside as there are negative ions inside a cell; it's the very minute imbalance (fractions of a %) that gives a potential of ~-70mV.