Your agent should have advised a due diligence period to review info on leases, security deposits, rent history, etc., and that one unit would be vacant by closing if using an owner occupant loan.

My brother and I (who both have history of doing multiple live-in flips over the years) recently consulted with an attorney, started an LLC, purchased the BP state-specific forms, and have purchased one single-family rental thus far and are closing on a second singe family property that will be a flip later this month.

If you have performance history that is tangible, that helps too. 

But at least you can check under "Price/Tax History" and sometimes see previous rent listings.  

Verifying last 2 years of rental history very important!

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 applicant(s) involve a Father, Mother, 31 year old daughter and 7 year old son.  

Are there any free public sites to verify criminal and rental history in california?

Brief history.