Is the family sinful for leaving the mother to work alone at home, despite being employed, without help from the father or the children?

Not helping your mother with household chores is a shortcoming in your dutifulness towards her. If she asks for help and you do not respond, that is disobedience and undutifulness ( 'uquq) for which repentance is obligatory, because obedience to parents is obligatory unless they command disobedience. The mother is owed twice the dutifulness owed to the father. You must help her, and it is not permissible to belittle her, for she is volunteering her service to you, as she is not obligated to serve her children. Similarly, a husband should help his wife in what she does, following the example of the Prophet, peace and blessings be upon him.