We also go over the Leases in person with the tenants and explain that the 2nd offense is a mandatory court appearance and a $185 fine.

They do criminal, eviction, and how much they make compared to your rent.

But if everything else checks out for credit, criminal background, evictions, etc, but their place is filthy, how can you legally deny them if they meet the other criteria? 

I explained the rules on paper, have a very detailed lease, screened for criminal, credit, and income, and am very handy.

No offense my friend, but your first step needs to be getting pre-approved with a lender before doing analysis paralysis. 

The cops won’t remove them since it sounds like they were living there with the permission of your tenant, so it’s not a criminal matter.

It's an interesting perspective From the report:"But it also is true that Morris does not have to return to the country to defend himself if he has an attorney here representing him.However, if he simply tried to ignore a lawsuit and not engage local counsel, she said the plaintiff could win a default judgment against him.And because these are civil cases, rather than criminal, a court would not take action to extradite a defendant to bring them to trial"

Adding to that is the younger generations (no offense meant) are fluid. 

I tend to base mine on the following:no bankruptcy in the past 5 years no foreclosures in the past 5 years at least a 530 credit score but if they have shown a good payment history for at least a year then I am not hard and fast on thisno criminal record no evictions current on all payments monthly income = three times the rent no utilities collections in the past 5 yearsno collections in the last yeardebt to income ratio is less than 40%You can add modify or do what you like with this list and for me it is a guideline to keep my emotions at bay.

I have found that when dealing with problems where a tenant is doing something offensive,  the best course of action is to be direct and matter of fact.