Changes of permeability and driving force during voltage clamp
Relative ion permeabilities determine the equilibrium for a given configuration of ion channels. You can easily understand what is about to happen to the voltage of a cell if you know the permeabilities: if the equilibrium voltage is more positive than the present voltage, then the voltage will increase over time. The rate at which it increases depends on the membrane time constant, which depends on the capacitance of the cell (which is based mostly on membrane area, and stays relatively constant) and the resistance; the resistance is, again, based on permeability.

So permeabilities tell you A) Where you are going, and B) How fast you are going there. This is all relevant if you are living in "voltage land": you can actually ignore currents completely and just think in terms of voltage. This is the situation in 'current clamp' recording and for the most part I would say is the easier way to understand the 'big picture' for a neuron.

If, however, you want to think about currents themselves, then you typically clamp the voltage. Clamping the voltage also keeps the driving force constant (at least over the time scales on which voltage-clamp experiments make sense). In that case, you are right, you can't ignore driving force, however, in that case you are usually setting the driving force (by setting the voltage), and measuring the current. "Driving force" is then only necessary to figure out the permeability.

If you voltage clamp a cell at a depolarized potential for a long time (minutes), eventually yes the ion concentrations will change. Voltage-gated sodium channels tend to inactivate pretty quickly, so you actually won't get much sodium current when you voltage clamp at -20 mV for a long time, at least not in the soma, but you might get more current out in the distal processes that are not clamped well and other ions like calcium. Cells do not hold up well under long-term voltage clamp, however. I believe the main culprit is calcium toxicity, though it might vary by cell type and other particulars of the recording configuration. You can also kill a cell by just disrupting the membrane and making it more leaky: the concentrations change faster than the pumps can keep up with and you get excitotoxic cell death.

I usually recommend Purves' Neuroscience as a basic textbook, and the Methods in Enzymology volume on Ion Channels edited by Rudy and Iverson for patch clamp techniques and related practical wisdom.