Asthma and Allergies as far as the relation. Allergies tend to trigger histamine to be released which can cause swelling, rashes, itching, etc. The swelling can have the side effect of fluid accumulation from mucus membranes. Asthma is a constriction of tiny muscles deep in the lungs, as they constrict it takes more time and force to breath through the smaller tubes. an allergy can cause swelling of the tissue in the upper and lower airways having the same affect as asthma but also fluid accumulation. In a severe allergic reaction you can have constriction of the upper airway. Any fluid, swelling, itching, constriction of the airway can cause coughing. As far as an allergy causing or triggering asthma, I am not sure.
Keep searching the net for information and help, try sites devoted to asthma and allergies. also go to sites with message boards. you may find other people there that may share signs/symptoms, and find information on some specialists that may be able to help. I am sure the answer and help is out there somewhere. as far as going out west for TB, I have heard of many people with the condition moving to places like Arizona to improve their conditions. make sure it is a warm dry environment with CLEAN air. But remember if it is an allergy problem the pollen and dust flying in the air in these areas can cause problems as well. What ever your sister's underlying cause you should get a high quality air filter for the living space. She is also going to need to understand what things she is allergic to and do her best to avoid those things the best she can.