Oakland bans criminal background checks on potential tenantshttps://www.latimes.com/california/story/2020-01-21/oakland-bans-criminal-background-checks-on-potential-tenants1.

If your property manager has failed to remit funds they collected that is a crime and you should file criminal charges against their principals with the court commissioner. 

Things I look for, and include in my listing: no evictions or major criminal background issues, take home income at least 3x monthly rent (this is personal requirement, and higher than the typical rent = 30% of gross).  

SEC 8 ONLY screens for criminal. 

And if they apply and pay the cost I run their credit and criminal. 

I have the income requirement on the application now, but I'm curious if other PMs list things such as minimum credit score, eviction-free period, and criminal check history requirements.

No criminal history (or whatever level you are comfortable with)Speak with past landlords, employers and have a background check completed. 

The Dept. of Justice says rebates are allowed in Arizona and 39 other states. https://www.justice.gov/atr/competition-and-real-estate-state-7 But the Dept. of Justice website calculation on Realtor Rebate Savings is very weak as you can do a lot better  https://www.justice.gov/atr/consumers-save-thousands-commissions Also, depending on the loan and type of property you're purchasing, Buyer Agent Rebates can:pay for Interest Rate Reduction pay for rental car for buyers while in FLpay for plane tickets to FL to shop pay to eliminate PMI pay to buy out remainder of leasepay towards down paymentpay for the 1st full year of home ins. pay for taxes and ins. escrowspay for a Home Warrantypay for home repairspay off credit card debt, auto loans, etc.pay off collections and judgmentspay for new furniturepay for all new appliancespay for home remodeling and additionspay to have home painted pay for weekly pool maintenance pay for credit repair pay for landscapingpay for monthly lawn carepay for new A/C system...........and pay portion to prior agent for their efforts

You are not liable if you unknowingly rent to a criminal. 

Anything above that is just icing on the cake if it happens.What the person wanting to live in one unit and rent out the other two has to answer is:1.