What is the ruling on dealing with the non-Muslim children of one's spouse who refuse to embrace Islam and live the life of disbelievers, and do they have a right to inheritance?

Your husband's children have no right to inherit from him due to the difference in religion. They are your mahrams, and it is permissible for them to look at you, with a difference of opinion regarding the limits of what may be seen. You must be diligent in inviting them to Islam and clarifying to them the danger of disbelief.