Is it permissible for a woman to remain with a husband who neglects prayers and sleeps through them, while she fears for her faith and her children's faith from being influenced by him?

We advise you to supplicate frequently for your husband, especially during times when prayers are answered. If your husband is negligent in his prayers, to the extent that he offers them outside their prescribed times, then a severe warning is due to him. So, continue to advise him and remind him of Allah. If he repents, then all praise is due to Allah. Otherwise, it is commendable for you to separate from him and seek a divorce. However, if he sleeps but does not let the prayer time expire, then there is no sin upon him. Nevertheless, it is not permissible to sleep after the prayer time has begun for one who knows that he will not wake up before the time expires, unless he has entrusted someone to wake him up. As for backbiting people and disobeying one’s mother, these are major sins. So, advise him and remind him of the grave consequences of these actions.