Is what the husband's family is doing considered a breach of the prior agreement to reside in the city? What is the Sharia ruling on what the wife and the husband's family are doing? Is the wife's service to her mother-in-law obligatory?

If you stipulated to your husband that you would live in the city, then this is a valid condition that he is obligated to fulfill. It is your right upon him that he provides you with an independent and suitable dwelling. You are not obligated to live with his parents or others, nor are you obligated to stay overnight with them. They do not have the right to take your child to stay overnight with them, and serving the husband's mother is not an obligation upon the wife.