I don’t believe the seller or listing agent is at fault as you can rent in the community (with restrictions)it is up to the buyer and buyer only to do the due diligence in these types of matters.Similar to ones that say pets allowed - doesn’t mean you can have a lion.

I prefer no pets.

It is not harmful to pets or children either.

Stated: Previous addresses on application match previous addresses on credit report, application filled out completely/correctly, specify # of instances and how long ago criminal activity needs to be, no bankruptcy in past 3 years, not turned into collections (you effectively have this), no evictions.Non published things to look at: smoker, pets, has bank account, has lived at previous addresses for over 1 year, roots in the community, not fussy, how long indicates plans to stay, how well their desired move in date aligns to your date, has sued or been sued.

But while they are there, they look around for anything else that may need to be done and also just over all check on the property and if tenant is obeying rules (like pets for example). 

Zillow sends an email with the person name and profile info (if they filled it out), which has name, number, income, pets, etc.

As many tenants will tell you to your face that they do not have a cat or dog, and then try to sneak them in...What about the following idea: Ad text say "pet friendly" Seems people are more likely to reveal pet ownership that if you say "no dogs/cats allowed."

You can (and should) make them pay full security deposit, any pet fees (but probably no pet rent, unless you are under the Sec 8 rent) and do full credit and background checks (there will be debt, maybe collections, but you can screen some criminal, depending on where you are, and evictions -- again, depending on where you are).Let's say you are charging $800. 3 times is $2400/month.

I often allow pets and have had some great tenants with pets, but have occasionally been left with smelly apartments.

I also think you are parroting some things you've read in books or heard at seminars that rarely, in reality, work out.