In China, do they have "Pot Luck" dinners where many families get together and they each bring a a dish of food and then share it?

No really unless in certain occasions like funeral, sickness, etc that kind of situation where the host is not capable of doing all the cooking. Chinese culture is all about "saving face" or avoid humiliation but to show off most of the time especially in invitation to their birthdays, Chinese New Year...meaning getting together with families and close friends in a private gathering and not in the office parties, picnics, or casual kind of settings. In private celebrations, the Chinese culture provides everything and if you were to bring your own cooking, you are not showing friendliness but will offend the one who is cooking like to a challenge your food is better than hers, or the family does not provide enough may be you are looking down at them, or simply being defiant and even rude to bring your food. Unless, there is a funeral, sickness etc and the one who is doing the usual cooking is not able to due to grieving, mourning etc. then yours and others contribution to feed those who visit to that family, will be appreciated. If there is a community, public, festivals etc celebration, then its time to show off your best recipe.

Yes, its very common in Asian culture. Even when Chinese people eat together they usually will order many dishes and then split the food instead of just ordering for themselves.