-I’m most interested in their income, verifying their credit and income matches what they told me, and of course no criminal/eviction history.

I would also suggest making sure that the criminal background check comes back clean, since the screening recommendation states that it is only based on credit.

I notice that most of the examples of references "saving" someone from making a bad decision on a tenant involve instances in which the problem would have been detected with a simple background, criminal, or credit check.  

Criminal background: No violent felonies within the last 7 years.

You can reject a Section 8 applicant for the same reasons you reject other applicants: credit score, Landlord reference (or lack thereof), criminal history, etc.

.: Sorry, but 550 credit rating is for criminals.

You can still disqualify them for any reason you would normally disqualify someone (credit, criminal, lack of references, etc.).

Ex: We just had a resident apply to one of our buildings who at 20 has a criminal record 12 pages long. 

Run same background/criminal checks on them as you would anyone else.

All roads closed, check points, criminal to be out of your home.