In case of a suit, will insurance companies or courts require a DNA test if a dog is suspected of being something other than what is listed on the vet records?
I'm not sure there's a hard and fast rule on that. It will depend on the insurance carrier, the judge, and the circumstances.
As a landlord or property manager, the best way to help protect yourself is to have standardized pet screening policies/procedures and document the level of due diligence. There are companies now that will do this for you (such as www.petscreening.com), on an outsourced basis. They also vet service and emotional support animals, and even provide a "no pet affidavit" that sets expectations for tenants who claim that have no pets.
We have no affiliation with PetScreening.com, but we do use their services and are generally very happy with the process.
For pets, they collect photographs of the pet, as well as information on vaccination records and behavioral/bite history.
For service animals, they verify the prescription with the healthcare provider.
If nothing else, we can genuinely state (and document) that all applicants/tenants with pets, service animals, and/or emotional support animals are screened the same way and treated exactly the same based on a standardized set of industry-accepted criteria.