reddaisy - have you ever had a cat? I'm quite allergic to them but when I have my own it clears up after a while as long as I don't rub my eyes (a bit of a hassle, but not a major drawback) and I only get inflamed skin if I get scratched.
'Small animals' really aren't good for small children as they're too easily hurt or killed :(
Pestering is irritating, but hey, it's just part of having kids isn't it. I'd make sure that she knows you have heard her, be very clear and tell her that you understand how much she would really like a cat, but that you are allergic to them and it wont be happening, no matter how many times she asks. Tell her how sore it makes your eyes etc and ask her if she would really want you to hurt like that every day?!
If she's the sort that can be 'jollied' out of things when she says 'Mummy I really want a cat, can we please get a cat' just say 'Oh I really want a camel, can we please get a camel' - just joke about with her - but not if it will make her angry/upset!
It took me until I was 14, but I finally got my dog, only about 10 years of pestering!! 