When its sold, can the funds be use to pay off their current primary residence with a 1031 exchange, or can that money only be used towards the purchase of new properties? 

You can do anything from traditional mutual funds and stocks to buying real estate and businesses with your money.

Going through a traditional bank can be a tedious process and no guarantee that you'll even get approved for a loan.

This is my primary goal ..kind of hitting the reset button on life. 

I make up for it by having higher amount of liquidity and alternate income (outside of my paycheck) vs something tied up in a property that is my primary residence....1.2 is a good chunk outside of CA... you'd typically see something around 1-1.5% rents to purchase price and at the price point you'd be able to get a portfolio of multiple properties.  

I am searching for a new loan, I must refinance my primary home to get out of a horrible interest rate. 

Well, this did not work out since i was getting a primary residence home loan.

I already have a HELOC on it and I am trying to raise my HELOC balance but the issue that I am running into is that the larger institutional banks are not willing to use the rental incomes from my 3 units toward my income in determining my debt to income ratio because the building is my primary residence. 

I have only purchased my primary residence, and this is a new learning experience for me.My question is - what are some of the real life pros and cons of using a Hard Money Lender over an Equity Partner?

The gain related to a primary residance portion of the duplex is sheltered via section 121 gain exclusion.