As @Frank M. pointed out, you'd need to have a cash reserve in your IRA in the event of an unexpected repair, or vacancy that limits or stops income generation.

If you underestimated repairs or didn't set aside sufficient reserves, that is not your tenants fault. 

The problem is she does not have enough cash reserves for a down payment on a new smaller home to purchase in order to rent our her current home because the prices are so high in Orange County right now. 

You can BOTH use the same checking/savings/reserve account to show reserves (which get pretty hefty as a requirement when you add from 5-10) AND for down payments. 

This would give me the $28,750 I need for a down Payment on the new rental with $10,250 spare as reserves for 6+ months on the property and any light repair work that may be needed.

I suggest you try to use Freddie Mac form 998 or Fannie Mae form 216 to calculate your reserve fund needs - see link below.

We reserve the right to turn tenants down for criminal history but this should really reflect a reasoning as to why.

Just be mindful that although they advertise 10% down, they also require some additional capital in reserves, so you need to show around 20% to qualify.  

:\The lender that didn't know you couldn't use gift funds for investment props probably also has that retirement account tagged to satisfy the "PITI reserve" requirements.

-I have money set aside in an account for reserves which I can show the investor.