Am I obligated to house and care for my sister, who suffers from schizophrenia, along with my wife, knowing that my wife is distressed by her behavior and desires independent housing for her, and will I be held accountable before Allah if I do not house and care for her?

Your sister has a right over you regarding maintaining ties and kindness. Her care is obligatory if she suffers from schizophrenia and has no one to look after her. Her financial support becomes obligatory upon you if she is poor and you are affluent, and this support includes providing service, medical treatment, and care. You should either house her with you or close to you, while ensuring a separate dwelling for your wife, meaning she should have a portion of the house with her private facilities.