What would the two graphs look like if switched from a, to b, then back to a (instead of c)?
With the switch in position B, the current in the circuit will climb to the maximum value, as limited by the resistor. If you were at this point to open the switch, and if all these components were ideal, the universe would explode. Whatever current was flowing in the inductor must continue to flow, but there being no path for it to do so, we have created an impossible situation which can only be resolved by the destruction of the universe.
In reality, the current would probably continue to flow through an arc in the switch, until there is no energy left in the inductor. It's hard to say exactly what the graphs would look like without knowing exactly how the switch will arc.
Why is a resistor used in the circuit? Alternatively, what would the graph look like without it?
The resistor serves to limit the maximum current in the circuit. If you remove the resistor, then when the switch is moved from A to B, the current will grow, linearly, forever, and the voltage over the inductor will always be the supply voltage.
In reality, at some point your voltage source (battery, in this diagram) won't be able to drive enough current to keep this going, and the current will stop growing, and the voltage will sag.
Could the first voltage spike be taken care of with a voltage regulator and the second with a diode, or would both be handled by the regulator?
I'm not exactly sure what you are asking here. The 2nd graph, the one in green, is showing the voltage over the inductor, not the voltage across the battery. I'm not sure how one would "take care" of it.
The easiest way to understand inductors, I think, is to realize that they are the dual of capacitors. So, if you take everything you know about capacitors, but exchange "current" with "voltage", and "series" with "parallel", basically everything holds true.
You can also consider them as a sort of flywheel driven by current. They resist changes in current, just as a capacitior resists changes in voltage. Current through an inductor can not change instantly, just as the voltage across a capacitor can not change instantly. The rate of change of current in an inductor is proportional to the voltage across it, just as the rate of change of voltage in a capacitor is proportional to the current through it.