The part of the agency that purchases properties may have investment criteria that certain properties do not support, so they sell them because they may meet another investor's criteria.   

-you likely won't cash flow at all when you refinance, and you might even be negative. 

I have spent the last 8 years co-managing a small remodeling LLC, and have an primary background in project management, client coordination and admin communications/support. 

A friend/relative that has either cash or equity that would lend it to for 6-8%?

First a cash snowball approach to build up your cash.  

. ~1.1 M in cash).What is the best time to buy the replacement property?

Once I built up enough cash, then I became a buy and hold landlord of single-family homes. 

So, with no foundational knowledge of real property, and no capital, his only alternatives are (1) “bluff” the homeowner into thinking he has the money and intent to close the purchase himself and or convince the homeowner he has an extensive list of cash ready interested buyers (“partners”) with whom he’s done dozens of prior deals or (2) pack up shop, write off the money invested in mentoring to a worthless investment, and decide IF real estate is right for home.  

Cash out refis ok, but may result in slightly lower loan sizing. 

In your shoes, it might be simpler to stay focused on grad school and use your cash to lend to other investors or take an LP role.