In a house that old, your looking at potential for inhaling lead from old lead paint (can become toxic), asbestos (cancer causing) and mold can really deteriorate your health in a big way. It does not sound like a very healthy environment. I'd move if this is a constant problem. Be sure to limit the dust by blocking the lips under your doors that are in contact with the dust areas, and change your air filters. Dust and clean frequently as well. You need to tell your landlord about the mold and have them fix it right away.
The plaster dust isn't good for you b/c of your COPD. It can irritate your bronchial tubes and your lungs, possibly leading to an exacerbation. I do think it's a concern, but not nearly as much as the mold in your basement. Mold in a house is bad, bad news. My husband works in property management, and his company routinely moves tennants and tears out any drywall once they've found mold. The workers all have to wear breathing masks because the mold is so dangerous. If the mold is black, it can be toxic to anyone, not just those with COPD. I would call in someone to look at the mold and stay out of the basement until the extent of the mold can be assessed. I would also let your doctor know that you are dealing with these conditions in your house. He can give your an expert opinion on how dangerous it is and what you can do. Good Luck!