I have the same reactions to most animals. The whole itchy eyes and skin. But I tend to adjust and adapt to animals and eventually my symptoms go away. The whole chest hurting aspect I am not quite SURE my answer will help you because I am an asthmatic. These are my personal ways that I get over chest is drink warm water, relax (I know it sounds odd but if I sit up and just work on slowly breaking it helps relax my chest till I can breath properly) also IF it is the cat....wash your hands face and drink warm water. That tends to fix my reactions even to those extremes. But I only get THAT bad if its a very shedding, long haired feline, and unclean environment (like not vacuuming for a week and such with cat hair all over) but yeah sometimes my reactions do go extreme
I have read that some reactions come from the saliva of the cat from its grooming itself. Can always try bathing it, grooming it often, cleaning the house more(not saying you are messy but the less dust and hair the better right!), keeping it out of the bedroom, or well like me waiting it out.
I hope this helps you a little bit. at least with dealing with the heavy chest at night till you find your personal solution. You might wanna consult a doc either way, Be it an allergic reaction the feline or something more serious, its always good to know.
http://en.wikipedia.org/wiki/Cat_allergy
your question got me curious and some info in here about it might help you.