I have allergies (dogs, cats, dust mites, trees, grass...you name it) and asthma. What works for me is no carpeting, making sure the animals are all brushed daily (helps a LOT with the shedding), and meds. I take Pulmicort for asthma, and while I *should* be on a daily allergy medication, I opt to take it only during the "bad times" of the year (spring, fall, mostly).
I've ALWAYS had pets, and I've ALWAYS been allergic, and have had asthma since I was 12. Honestly, I think for *me* it's a matter of controlling the symptoms so that I can enjoy my furbabies. I can't IMAGINE a house without a couple of cats and a Boxer now! Yes I have my "bad days", but for the most part, I'm perfectly content.
Good luck - you've gotten a LOT of great ideas here.