Is it permissible to live in the house of a person who does not pray and consumes alcohol, knowing that the dweller prays and observes the rituals of his religion, and the dweller is compelled to do so?

It is not permissible for you to live with your relative's non-mahram wife, due to the Shariah prohibitions that would result, such as khalwah (unlawful seclusion), and the ruling does not change based on her age or beauty. As for your relative's abandonment of prayer and consumption of alcohol, these are grave major sins. The best way to deal with him is to persistently advise him and be patient with him as long as there is hope that the advice will benefit him, for that is an invitation to goodness and an act of righteousness. However, if he insists on abandoning prayer and the advice does not benefit him, then he must be shunned and kept away from.