Wax will not hurt the emblems, however, there might be white specks which you'll need wooden toothpicks to remove.
You generally only wax the painted portions of the vehicle. However, it shouldn't leave streaks or anything if you do wax them for some reason. I did it to my dad's Corolla when I was a little kid :-p
If you got wax on the lights, that's not a problem at all. Just let it dry to a haze and wipe off.
Best way to remove the wax from your hands is to use dish detergent, especially if you hands are beading water.

You can try to use rubber or latex gloves in the future if you desire.
Wax isn't really dangerous for you unless you try to eat it or something - which I don't forsee you trying to do. Wax is not harmful to any surfaces on your car, except perhaps for some rough/textured plastic parts because it is very difficult to remove.