There's a lot of red flags on this and I would personally move on.If you still want to pursue it after looking at their background and credit report, might I suggest  a higher security deposit or pet fee up front.Still, there's always better tenants out there for you.

Ya I hate hate hate those people selling "secrets" or trying to recruit on here its one of my biggest pet peeves. 

Best of all it's nontoxic and won't harm pets. 

I'm generally in the No Pet side, but the yard lends itself to outside dogs (fenced in plus fenced kennel).

I think we get slightly above market rents, due to allowing pets, being flexible with credit score, and marketing on Zillow, HotPads, type of sites more than Craig's list.

Gold to hug and love and stroke and pet and I will call him George!

You can also deny on habitability limitations (# of people vs # of bedrooms), a consistent no pet policy if they have a pet,  Megan's Law registrants, ex convicts on parole or Sec 8 applications if you do not want to do the work they request after their inspection.

Everything I've read about changing the terms of a lease has to do with raising the rent, or no longer allowing pets, etc. and hasn't really helped giving me a real sense of what to do on this situation. 

My biggest or most unusual problems were: abandoned vehicles, a person escaped from a mental institution, tenants with pet rattlesnakes that escaped their container, a round trip visit to the dentist for a wisdom tooth removal, a one way bus ticket from Kansas to Georgia, cashing $25,000 worth of checks each week for a group of guys from out of state, etc.  

Screens happen to be one of my pet peeves, so I add them to my own annual maintenance inspection checklist.