Rule of Professional Conduct 1.1 requires that a lawyer provide "competent" representation. All mental illnesses do not render a person incompetent. However, if this attorney's mental illness is so grave as to render her incompetent, an investigation by the Bar should be initiated. If the questioner can substantiate some of the claims made in this question, I suggest he/she file a grievance with the Bar Association.
The question of whether or not a New York attorney should be allowed to continue to practice law is determined by the Grievance Committees of each of the Appellate Divisions of the Supreme Court. Information on the Grievance Committee for your area (Nassau and Suffolk County) is available at the link below.
Whether or not an attorney has an illness -- mental or physical -- that renders him or her incapable of carrying out his or her duties as an attorney, is a matter that should be brought to the attention of the Grievance Committee. I would add that a layman's diagnosis of mental illness may not be accurate. Many lawyers have quirks and eccentricities. The essential question is whether those quirks and eccentricities prevent the attorney from functioning, and whether or not they cause the attorney to violate the code of professional responsibility.
An attorney with a mental illness should be allowed to practice law as long as the attorney was competent to practice. If the illness was pervasive enough that the attorney could not longer competently represent clients, the attorney should cease practice until the illness dissipates.
An attorney can only ehtically advertise if she is licensed to practice law. If the attorney is incapacitated and has lost her license or had her license temporarily suspended, she would not be able to advertise as a licensed attorney.