Dates: Go by who the invitation is addressed to. If the inside envelope says "James and Guest" then you are welcome to bring a guest. If the envelope just says "James" then you are the only one invited.
Children: Acceptable to bring only if they are specifically invited. (Again, look to the invitation. If it says "James and Family" or "James and Mrs. James and Kid 1 and Kid 2" bring them along. If they are not mentioned, get a babysitter.