You will be assigned to the same table each night and for the same time frame. Your kids should wear the same types of clothes an adult would. Just no shorts, T-shirts or tanks. Nice slacks and nice shirt would be OK for casual nights if you are tqalking about the main dining room. If it's buffet - nearly anything goes (no swim suits!)
Not sure how many nights they put the kids together to eat. Try going on Royalcaribbean.com on the FAQ and this might help -r or give the customer care a call.