Is it permissible for a woman to sleep with her husband in one bed while knowing that he commits adultery, does not show protective jealousy over his home, and encourages her to commit unlawful acts?

What you have mentioned about your husband indicates his weak religious commitment, a deficiency in his manliness, and poor character. However, as long as he is a Muslim, your marriage to him is not annulled by his engaging in prohibited acts or by his injustice towards you. Your sleeping with him in the bed is not forbidden; in fact, you must answer him if he calls you to bed, unless you have a valid excuse. But if he does not repent from these prohibitions and treat you with kindness, then it is better for you to leave him, either through divorce or khul'. Indeed, some scholars have even gone so far as to say that it is obligatory to separate from such a husband.