(even still, we made the decision to allow pets) So, I guess I didn't think we had to allow just anyone in this space, with other guests right above them in the same house.

So far, the only red flag is the cat they’d like to bring (and I don’t really allow pets), but it’s the mothers emotional support animal.

Insurance services, eviction protection, skip protection, bed bug/mold protection, loss of rents, pet damage protection, entity structuring and asset protection (from a consulting standpoint), renters insurance, tenant liability insurance, maintenance services to tenants and other PM's, tenant placement guarantees, "enhanced" monthly reporting, setting up entities, bank accounts, eins for foreign owners, utility management, etc.Take anything that is standard in your market and see how you can kick it up a notch.

How do I make sure that everyone is responsible for the damage - even if it was caused by one tenant/pet ?

They had a pet (Cat) last year 2014-2015 and not anymore .Tenant request the pet deposit refunded before they sign the new lease . 

That's fair and reasonable to me.Meet with the prospective tenants face to face when possible and ask questions related to their work, their past living circumstances, their pets, their vehicles, questions that make you feel comfortable that they will take care of your property and pay the rent on time.See the post about Craigslist in this forum for ideas about tenant locating websites.

Most tenants appreciate the privacy and how it can make it more kid/pet friendly.

If I own a duplex, rented one side and lived on the other, and the lease stated no pets (dogs specifically), can I own a dog/pet in my unit? 

State "pet friendly" instead of "pets allowed with approval" -obviously you'll have to approve them, but sound more inviting up front.

Agreed..everyone has different "pet" issues.