I am sure this is a frustrating situation for all of you. Actually, there are a couple of different answers to this. First of all, stopping the smoking is a good thing. It will reduce the cough and help the wheezing a lot. As far as the cats and the hamsters, it really has to do with your husband being allergic or not. If he is allergic, then they will definitely cause a problem sooner or later. If he is not allergic, as you are not, it will make little difference. Since this is a significant question, it might be worth seeing a Board Certified Allergist for one visit to get this question answered. During this visit the allergist could get the asthma medicines for your husband straightened out, as well.
I am Asthmatic and Have smoked, I also got worse since starting to smoke - because it was complicated with some bad bouts of bronchitis, i have two very hairy dogs (especially this summer)
and the hairs are flying around the house, quicker than I can hoover them up, but I never had an allery to animal hair and I still don't my asthma got worse because of smoking, moving to a new home in a new environment and my asthma still disimproves with changes in weather and dust. There are my triggers not that everyone is the same. It could be your partner got a bad bronchial attack which made his asthma worse, he could be left open to more allergens hence the cat hair, infact my GP gave out to me when I said I had dogs, but there was NO way I was getting rid of them. Keep the house well dusted and hoovered - remove all carpets and get in wooden floors, always wet down the dust and keep the mattress on the bed clean from mites. ( you both can do all this - he can wear a mask!!) One other option, go for a break somewhere local without the cats for a week or two - and see if the breathing is better, travel can interfere and make breathing worse so just somewhere local. Even if it is the case that he has an allergy to the pets, you can work out a way to keep them away from him as much as you can, I think my Asthma has improved because I have built up an immunity to my dogs dander, but regardless of what anyone says and I know because I am my own worst enemy - Smoking is not good - even at times when you think it helps you breathe better - it is definetly adding to his problem.
The cats are safe
Regards
j