HELOCs typically function exactly like a credit card except thy are secured by the property you are borrowing against, whereas a normal credit card is unsecured.

They may do it with just a lower LTV 60-70%.If they wont give you the HELOC I dont see why they wouldnt just let you open up a secured line of credit with the duplex being the collateral

I am now a retail real estate agent (and I privately manage SFRs for owners), the trustee sales are almost over for the mid sized flippers around here ("my" money is buying non-performing loans, better returns on time and money). 

I would also suggest studying the zoning code to determine if the city performs annual inspections. 

Your wisest course of action is to see if you could have the keys to your new place a couple of days early (after you have paid 1st months rent and security deposit) to allow you time to move.

Depending on the disclosures the buyer made for the sale of your home there might not be anything wrong .Perhaps the contract had an assignable clause or he advertised will be sold after closing.The agent might appear to not  be ethical but is simply selling a home for a higher price that is under contract  The fact that he did not cancel your contract might allow you to keep the emd or demand the buyer to perform  

I invest in notes and for my non performing notes there is a decent amount we end up foreclosing on.

I am curious to how the security deposit gets transferred when inheriting a tenant. 

The best way in securing a partner is to provide value to them, or prove you have the knowledge or expertise that makes it worth their time to "get in bed" with you. 

Raising money isn’t a do-it-yourself project, you need the advice and guidance of legal counsel that specializes in securities law.