For example, I mention that every adult must pay a $30 application fee, pass my credit/criminal background, make a combined income of 3x the rent, and whether or not the property accepts pets.

Are you willing to accept that?

Most lenders will only accept properties C3 or better. 

I've tested out some of them as well.

We go down to a 75k value/purchase price for example and some of the same tests that a loan this size may fail for a standard mortgage would not be an issue. 

I have a replacement property picked out already and an accepted offer on it.

I would get a water quality test though, included in your home inspection.

However, the agreement to accept monthly payments may be specific to the owner himself.  

If you just hire an agent, most of them are going to charge a one-month Leasing Fee because they offer 50% of it to an agent that brings them an acceptable tenant. 

But, due to being self-employed, it is extremely difficult to get accepted for any kind of FHA loan.