I don't think it matters, before or after. Maybe even both. With dry though, if you sprinkle it on top you might want to do it before so you don't leave yeast clinging to the top/sides of the fermenter.
Also, yes, calibrate your hydrometer. I don't think you have to use distilled, use whatever water you use to brew, it'll be fine. If it doesn't read zero, just figure out how high/low it reads and adjust all of your measurements by that much.
And, yes, get the beer cooler while it's fermenting. This is really important if you want it to taste good. Controlling your temps is huge. The water bath / frozen bottle method described above is an inexpensive and very effective way of doing it. Warmer ferments faster, but speed is not your goal.
Let us know how it goes.