I want to refinance this property and take out the money to buy another rental property (or at least put a good down payment for one or two).

Typical bank draw cycles are at least 30 days plus a few days for the bank to process.If your contract stated payment immediately upon completion then you have a solid argument.

As I do not make much to put a whole lot for down payments.

Myself and my Wife currently own a home (in my wife's name) that I make the payment on.

RoI I listed is after my payment and interest.

Some people look at cash flow as the renter giving you back your own money (down payment) a little at a time.

As an example, let's say the person is about to go to foreclosure because they got behind in their payments due to a job loss for 4 months, but have now found employment, but can't get the back payments caught up.  

It sounds like you are getting too soft maybe with rent payments.

We first reached out to originators for Freddie and Fannie and were told that the deal looked great and we should be able to get the following terms:-Non recourse loan, 80% ltv-30 year amortization-10 year fixed rate-4.25-4.5% interest rate-Interest only payments for the first two yearsFreddie has since come back and passed on the deal due to our market being too small and the age of the property (1984).

nope, these two actually have the documented ability to make the payments at a reasonable debt-to-income ratio.I'm coming up with ballpark $345k/mo PITI (my homeowner's insurance assumption might not be accurate, I didn't pull the actual property tax bill, and I don't know if there is an HOA so I left it at $0).