Kromey:
...Although water as the go-to analogy for electricity doesn't really work. For example, if I have a 12V wire and then I split it into two, I have 12V on both. Whereas if I have a 12GPM pipe and split it into two, I have two pipes that together add up to 12GPM.
That isn't so far off from voltage. I always imagine voltage as water pressure, and current as volume. Given the same amount of water, if you make the pipe bigger (more current flow), the pressure goes down (lower voltage). This frequently happens with electricity too. It's called load drop. The voltage regulator can compensate for this, just as a water pressure regulator could -- up to the point where more voltage/pressure doesn't exist at the source.
Kromey:
If I were using a transistor to drive the relay, rather than the Arduino's input pin being on the same line directly, does that in and of itself provide enough protection? Or is there still the risk of that spike "crashing through" the transistor and down the base lead?
Yes, you still need a diode. Afterall, the output pin of the AVR is just a pair of transistors.... 
polymorph:
You do NOT need a 1000V diode. The diode only needs to withstand the driving voltage. When the contacts open and the circuit is interrupted, the diode is forward biased and only sees its own forward bias voltage.
While I am willing to be wrong here (in fact, it would make life easier!), isn't the voltage rating of a diode relevant to the reverse bias? In fact, the Digikey sort heading is: "Voltage - DC Reverse (Vr) (Max)" If you spec the diode for 5vDC and it receives a reverse 100v spike, are you telling me that wouldn't cause damage to the diode? Then, frankly, why is the diode necessary at all? Couldn't the transistor indeed cope alone?
I don't argue that 1000v is probably excessive, I admitted to that myself above. It's just that 1000v ratings are not hard to come by, and it nearly completely eliminates any cause for concern or need to calculate margins. You can slap a cheap 1000v diode in there and worry not. If the spike exceeds 1kV, you have bigger problems. All IMHO, of course.
Also, I'm not entirely sure if you're referring to the diode in series with the I/O pin to the relay coil, or the one across the coil. One will experience a high reverse bias voltage, the other will see a high voltage that could be of either or both polarities.
polymorph:
The maximum current will be exactly what the relay draws when the switch is opened, no more. Then it decays rapidly.
When did we start talking about current? I agree, current demand will be low. Usually tens of mA.