Why is membrane potential not zero at equilibrium?
The value of the membrane potential does not affect the location of the resting potential.  The membrane potential affects only the fraction of channels that are open, thus allowing net current to change, which allows for the depolarization and repolarization of the membrane (the action potential).  During the action potential, the resting potential does not change.

The value of the resting potential is governed only by the distribution of ions (how many are on one side of the membrane vs. how many are on the other).  This is relatively constant even as ions cross the membrane for the action potential; it is really not a very large fraction of ions moving at all compared to the reservoir they are in, so the reversal potentials are preserved over time.