Can you develop new asthma triggers to things that have never bothered you before, such as things in your home?

New allergies occur: Asthma is an inflammatory disease of the lung airways and one of the major inflammatory mediators is allergy. Allergies are dynamic so if you have a tendency to be allergic you can develop new allergies. Therefore the answer to your question is yes.

Yes: Especially if your asthma is not controlled, you can get sensitized to multiple things that you are continuously exposed to. It is a vicious circle. You need to control your asthma and allergies and do your best to avoid triggers.