Xei, pure water is a very good insulator. It conducts electricity about as well as space conducts sound (that is to say not non-existent, but of no practical significance). I'd feel pretty safe jumping into a pool of distilled water holding on to a plugged-in toaster.

As for the rate of electrolysis... well try this experiment. Take a glass of tap water and put copper wire in it connected to each lead of a lantern battery, then do the same with a glass of the same water with a couple spoonfuls of table salt in it. You'll notice a lot more action in the second glass. If you can get a hold of some laboratory grade distilled water and try the same thing, you probably won't notice any reaction at all.