Summarize:

Traditionally, a dipping sauce of raw eggs is served alongside sukiyaki. The combination may not sound especially appetizing to westerners, but it is a surprisingly good combination, even if you aren't predisposed to liking it. If you are worried about uncooked eggs carrying salmonella or don't like the flavor, omit them. For what it's worth, salmonella is very rare in raw eggs. Simply grab whatever looks appetizing out of the hot pot and dip it into the egg. If you are not eating with family or close friends, use the thicker ends of your chopsticks to grab food out of communal pots. This is considered polite. If there are leftovers that didn't find into the first generation of the meal, cook them now in the remaining broth. Meat and leafy greens will take less time to cook than vegetables will. In Japan, it is traditional to use the leftover broth as the broth for udon noodles. If you don't have enough broth but wish to make more, use the recipe for sukiyaki sauce above and combine it with the remaining sauce. Heat the broth back up to boiling, then reduce the heat to a simmer. If the broth tastes too overpowering, cut the broth with 1/3 cup of water and taste again.
Prepare a dipping sauce by beating one egg for each person being served. Dip meat and vegetables into raw eggs before eating. Continue adding any extra meat and vegetables into the hot pot as you eat. If any broth is leftover, reserve it and serve udon noodles with it the next day.