People who have service dogs (or horses, as these are the only 2 service animals) that are genuine have no issue disclosing that on the front end and can answer the 2 questions easily - is the animal necessary for a disability, and what specific tasks has the dog (or horse, if you're getting one of those) been trained to perform? 

(Insert some sincere rapport-building compliments or questions here - about their yard or dog or whatever.)  

However, cats can cause far more damage than dogs.

Do any of the tenants have Dogs (Breed restrictions are individual to each company).These may not prevent you from getting insurance but they may limit the number of companies willing to insure it.

I have spent time on the PaperStac site, but they seem like mostly “dog” notes and not ones that I would want to own or trust to buy.

So you are not bound to "bird dog" for them. 

If the neighbor owns a dog or has other reason that even a temporary dismantling could be problematic, you'll want to let them know that the fence is being replaced so they can mitigate any damage to your property or crew while it is being replaced.Hope this helps...

Not sure if the upstairs tenant (heavyset person with a dog) plays a role.Seeking suggestions.

You have no more at risk than you do in your own home (where you probably have more equity.) or your own car (where you probably have more risk.) and we won’t talk about if you have a pool or a dog. 

FWIW I am only concerned about cleanliness when we rent...and that they allow dogs.