When I have people interview and ask, I reply we have 2 1/2 dogs and 1/2 cat. This of course gets a chuckle with a few raised eyebrows. I then explain:
We have 1 (outside only) dog who is a Collie. He's got very long hair, sheds a little, but is strictly an outside dog. If it's dangerously cold he'll spend the night in the garage, but so far this year that hasn't been necessary.
We have 1 (inside) dog who is a Mini-Schnauzer. She does NOT shed at all, and is great with the kids.
We have 1/2 (inside) dog who is also a Mini-Schnauzer. She is my husband's dog and is only home when he is. She spends the majority of the time in the co-driver seat of his semi-truck. She's his alarm system for the truck. Sounds funny, but trust me, if you saw that dog in action you wouldn't doubt me. When she's home she's great with the kids also. She's here 1-2 weeks a month, depending on the month.
We have 1/2 cat. Really she looks more like a raccoon than a cat. She has a name, but we typically call her "basement kitty". She has taken up residency in the basement. She HATES our two dogs. Our basement is off limits to daycare children (unless we're practicing for storm drills) due to lack of egress. They have heard her and seen her. Every once in a while I'll lock the one dog up and bring her upstairs. She sheds tons, so really I don't mind her staying downstairs. Less hair mess for me to clean up.
So since the kids rarely see her we consider that a half cat.
As for allergens, we have a friend who is severely allergic to cats, yet with her (the cat) staying downstairs he can come over and is not at all affected by her presence. The schnauzers have hair with hair follicles, not fur, so therefore they don't have pet dander. It's like people hair all over their body. Some other dogs are like this too, like poodles (and I'm not sure which other types).
As far as licensing goes I am required to keep my pets healthy and all of their shots up to date. This is done anyway, but they do check our records. I'm not sure if my county has any limit to the breed of dog, but I know our home-owner policy does.
Other pets - I'm not sure about reptiles, but birds would need to undergo certain testing to ensure they don't carry some disease. I don't think most fish area problem either, of course if I had a piranha tank that might raise some questions/concerns. I'm also not sure about small animals/rodents (hamsters, gerbils, etc).