I researched that extensively since our twins were preemies and everybody and their brother thought they should tell us how living in a house with multiple dogs and cats would cause them allergies and breathing difficulties.
Well, they came home after over a month in NICU (my daughter on a heart monitor), and from day one a dog has slept either in their room or in our room where they co-slept with us.
Today they are 8 years old, each with a dog in their bed at night, and they are healthy as can be. Zero allergies, asthma, breathing issues, nada. They actually get fewer colds and germies than their classmates.
There was one study done in England I think that showed that children exposed to a compound found in cow manure develop more of a protection against auto-immune diseases. Over sanitation is more damaging in the long run from everything I have read.
Oh, but I also made the effort to breastfeed even though they were preemies. I pumped for months and managed to produce enough for both of them up until 4 months at which point my daughter finally figured out how to nurse and of course was able to get much more that way. I think that makes a huge difference too.
No, I would not rehome my dogs either. I've had an aggressive dog in the house with them, and doors are a wonderful invention. If the dog was so aggressive to not be trusted with careful management, IMO the dog needs to be PTS and not pawned off on someone else to deal with. JMO though