based on the symptoms your daughter is exhibiting, i would strongly recommend taking her to see an allergist to see what she is allergic to and how severe it is. once you know what her condition is you can make informed decisions.
two things to keep in mind, avoidance is the least invasive option to reduce reactions to allergens. having a pet will increase both the pet's allergens and exterior ones (unless it is an inside pet). second item- allergies change, sometimes drastically, over the course of your life and vary by where you live (geographically).
my perspective is based on having had mostly untreated allergies as a child and teenager, which got worse in adulthood and in my 30's they got so bad i began immunotherapy (allergy shots). For me, the shots have been life changing and i now take almost no medication and have very few symptoms. i would not recommend shots for someone with less severe allergies tho.
i did try going to natural products, organics, and a whole host of other things prior to taking that step. unfortunately none of those worked for me (many natural products actually made things worse b/c i am allergic to so many plants).
regarding being healthy: altho allergies are related to your immune system, having them does not mean your daughter will be unhealthy or sickly. i have always had a very good immune system and was a very healthy child and adult.
medicines to try: i tried a lot of different products. There is no 'best' product out there for everyone, different ones work better for different people with different combinations of allergies and it is likely you will need some trial and error to find the best one. i personally liked zyrtec best. keep in mind that finding the best thing can take a while b/c some of the drugs need to build up to work, and the allergy conditions are constantly changing. i now use a prescription eyedrop and nasal medicine which i rarely need but are very effective (better than OTC)
if your daughters allergies turn out to be seasonal: you can track seasonal ones online and have a good idea when to take medication.
types of pets: each animal allergy is specific to the species. each animal is a different allergy- dogs, cats, rabbits, horses, sheep (wool), birds (feathers)etc. having one does not mean you will have another, or that it will be as severe. some breeds and types are better/worse than others, but unless your allergy to that animal is very minor, it is likely almost any animal of that kind will bother you.
If you want to go to a specialist: Northwest Asthma and Allergy Center has offices in Seattle and around the state. I go to Dr Virant (in sandpoint, near Children's Hospital) and i would recommend him. i feel he offers choices without trying to push things on you. (i dont know how he is w/ kids- so far my son doesn't show signs of allergies. there are a ton of kids that go there and they have a toy drawer in each office :)
Good luck to you both!