My kids alway have playdates with their friends. Normally if your child is over 6 years old, you do not have to stay unless you also want to socialize with the parents, in that case, usually the mom stays. In terms of what to bring, I usually bring fruit tray. Home made cookies or muffins, just make sure they are nut free. Caucasians normally do not prefer to have the kid stay over for lunch or dinner and they usually like to give a time frame. Unless you know the family well or they tell you it is a party, in that case, they would let you know ahead of time.