What documents do they require, what credit scores do they allow, how do they verify previous rental history, etc.?

Verifying last 2 years of rental history very important!

The loan application information is then verified during underwriting by income documents such as pay stubs or tax returns. 

I'd remised to tell you both of these accounts we had to verify with  2 small deposits that took days. 

Verify they are licensed and insured.

I'd verify with bank statements the income as well

If the tenant has a support animal, how to verify the validity of it?

The only issue would be having to verify income for the lease renewal if needed.

Be sure to verify!

Take for instance my B+ located quadraplex consisting of 2 Bedroom units in Philadelphia that rent for just shy of $3,000/m and compare my building to the nearly identical quadraplex building in a C location in Philadelphia were the rents are $1,200/m.