Congratulations on entering the 3 speed cult! The washer you're talking about is an anti-rotation washer - because the wheel is internally geared, it needs something to keep the whole axle from rotating in the dropouts, and, as such, it's completely necessary. Harris Cyclery has the best selection of 3 speed parts I've found, but you might find a crusty bike dude in your neighborhood who will just give you one. They tend to collect 3 spd hubs.

As far as noticing the gear change, it's totally not obvious until in the frame - the only thing that changes is how often the wheel rotates relative to the cog.

When I true up these wheels, I hit all nipples and spoke holes with regular oil (auto, Triflow, whatever) the day before and let it sit. If this doesn't work, you'll know it, as you'll start snapping spokes as you go around. If that's the case, you can try replacing the ones that die, but it's a little dicey as to how well the wheel will hold up if it's up to tension.