Thank you for asking your query.
It seems you are having an allergic reaction/hypersensitivity to dust and cat hair.
Best treatment here is avoidance. Although you can avoid cat hair, dust might not be possible. You can take anti allergic drugs like cetirizine atleast half an hour before the exposure.
But please note, dust is a composite of lot of components. So it might be some pollens or smoke which is causing the allergy. If its possible try to identify the particular allergen.
To make sputum thin you can take Mucolite twice or thrice day with your Doctors supervision. But please not this not recommended for long term use.
Its wise to take aXray and make sure the lungs are clear.
Hope this helps.