Try getting rid of your cat and see if the symptoms go away
Well, if everything didn't just start out of no where when you bought the cat and you've always had some sort of abnormal coughing or tickling and some mucus you could have a mild form of COPD. I'm currently going around suggesting it to people who are being told they have allergies when no allergy medicines work.. I was told this for the longest time and believed it, but I stumbled over this recently.
Oddly enough what makes me think I have it now is my year old habbit of smoking Ciggarettes (don't start doing this) I've always had persistant cough of some sort runny nose etc etc, in COPD it says your lungs produce too much mucus which in turn causes damage to your lungs (actually very simillar to Emphysema) but I know for a fact I've had this problem prior to smoking (8 years old I remember having cough attacks) personally I think Allergies can make it worse, and confuses the subject.
Some sites that may help you out
-http://www.nhlbi.nih.gov/health/public/lung/copd/
-http://www.nhlbi.nih.gov/health/dci/Diseases/Copd/Copd_WhatIs.html
http://www.nhlbi.nih.gov/health/dci/Diseases/Copd/Copd_Diagnosis.html
Ask about the Spirometry test located in the last link.
Of course, this seems like a recent problem to you the way you described it, the thing is I've seen it stated somewhere that this condition can be set off by a number of things, it seems like a triggered condition, although you can be born with it too, but its more rare to be born with it. I'm no doctor, I just was looking around at somethings and noticed the connection between my early troubles and the link to my smoking making it worse.