Hi, I can relate to this very much. My5 year old daughter is allergic to dogs. She gets a runny nose, watery eyes and breaks out in excems. we did not notice this at first, it did take a few weeks like you. Every person is born with a thresh hold for allergins,(dust, pets, perfume..anything) and everyones is diffrent. Once that thresh hold is met, the body can not fight it off and runny nose, cough, hives, rashes,athsma can all occur from this, and not always right away. you say she is around other animals before with no problem, but she was not living with them every day, that makes a big diffrence. Also, have you tried giving the puppy a bath? she could be allergic to the shampoo. what about flea collar, or drops?
try giving the puppy a bath in diffrent shampoo, I use baby shampoo. Also thr changing flea products if you are useing them. do not use flea collars, they can make kids and people sick. I do not even think they work that great anyway. Also take your daughter to the doctor and see about allergy meds, My daughter is on singular and does great with it.
I hope this helps you and good luck!