What is the ruling on utilizing a house and utilities designated for the father's work (water and electricity) for the family to permanently reside in, and what is the ruling on the children benefiting from that, and do they incur sin if they were the originators of the idea? And is the contract considered valid if this house was obtained from the state?

The ruling on what you asked about refers back to the conditions set by the employer. If the employer stipulated that the worker reside alone without anyone else, and prohibited the sharing of services such as electricity and water, then it is not permissible for him to violate that. However, if the employer allowed him to house his family or whomever he wished, then there is no harm upon him. And if the state grants ownership of the house to the worker, then it is permissible for him to accept that. For Muslims are bound by their conditions, and they must fulfill contracts.