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

Missing documents or the wrong version of the documents is usually the biggest reason for the permit being kicked back. 

We typically look for at least a 600 credit score with a 2.5-3x monthly rent. 

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

Good word @Bruce WoodruffThere is also a site that is similar to AIRDNA with a free version called "awning".

I do however, have an ITIN, 720 credit score, and a B1/B2 visa. 

I probably have a promo code I could give you if the BP one isn't good anymore but I used the free version for quite a while with success.It's also pretty decent at pushing listings out to some of the major rental sites.

The problem that I am running into is the current conventional mortgage options in my area for investment properties are 25% down (even with a 700+ credit score).

The property management put these tenants in even with a credit score of only around 500.

At the core of the OPs conversation what were really talking about is building a DIY version of VRBO/ABB.