First, if you use the '6' die and choose one of the other five actions, do you put the '6' die on your turn card or the die from the action category you choose to use the '6' die on?
You take the 6.
Second, when you are dealt the 6 staff cards are they free to use, or do you have to pay their fee in order to use them?
Generally you pay for them and play them with a 5 die action. But there are guest bonuses that can help you play them, too.
Third, when you use a staff card to gain food and/or drink, does it go to the kitchen, or can you put it directly on a guest card?
Any time you take food (for any reason) you can always put it directly on a guest card if there is one in your cafe that will take it.