That being said, I bid on one that I couldn't come up with the funds for, and since there was no deposit on hubzu for the first 48 hours they just put it back out to auction.  

We use our HELOC like a checking account and deposit all our income into it and since we pay next to nothing in interest we're seriously impacting our principal.

I got a pretty good deal, got $100 back from each month rent, full security deposit, and he paid full closing costs. 

We have 60 days to deposit the check on another eligible retirement program or I.R.A. and avoid paying taxes  on it otherwise we will be taxed and penalized. 

These are great passive investments that pay a 12 percent annual return with monthly direct deposits.  

How would you handle the deposit that was given when they signed the first lease?

Just a plain old boring lease with a security deposit made, move-in date on the lease to be about 3-4 weeks after you intend to close on your new purchase.We can use rental income to offset PITI on your old house, fixing your DTI.After that lease is up, evaluate the performance of your old home as a rental property, and either keep or sell it at that point.

Scenario 2: Partner matches my initial investment on a buy and hold, What should I offer them? 

At this time I had to pause but decided to return the security deposit (which I still have).  

Concerning your situation, I would ask for a double security deposit to cover the credit scores and allow them a few months to pay it.