I don't know that any of us Barbarian Americans really eats sushi "properly".... the best lesson I think your girls can learn is that if you are polite, even if you use your entree fork for your salad, if you do so with panache it doesn't matter.
Attitude, confidence, and poise are everything.
Those slices of ginger are pickled. I like to eat them between different pieces of sushi as a sort of palate cleanser. I know it seems natural, but putting ginger in your sake will NOT produce a sweet/hot ginger sake cocktail.
You can mix wasabi with your soy sauce or not, as you choose. I smear a bit on whatever pieces I want to. I've developed quite the addiction to the wasabi, and usually put enough on the simpler rolls (cucumber, california) to really vaporize the sinuses.
I eat small pieces (ones you can fit in your mouth) with chopsticks, and pick up larger pieces with my fingers and take bites out of them. On the one hand, the japanese think that taking a bite out of something is incredibly rude... on the other, they make those huge pieces that will NEVER fit in most mouths, so... <shrug>
Read the menu carefully, and ask if you have questions. Your first time out, getting mostly cooked rolls might be the way to go. Smoked salmon, anything baked or toasted, shrimp (usually... sweet shrimp are almost raw), tempura-cooked stuff, crabstick are all cooked ingredients. Most sushi chefs will take requests that aren't on a menu (particularly if they aren't busy), or can make suggestions per dietary restrictions or preference (cooked rolls, replace seaweed paper with soy sheet, etc).