It is just rate and terms refi, so we can increase the construction loan amount which will be taken on draws. 

In the case that tenants in all units have just re-signed 1-year leases, can one not use an FHA loan to purchase the property?

My initial thoughts would be to secure a construction loan for the total rehab amount based on the entire property's ARV.

I would like to report Xplicit Home Loan (https://www.xplicithomeloan.com/) for fraudulent activity.

There are a lot of details you haven't filled in in this post (which is fine), but you will likely need a key principal to co-sign loan and a contract with an experienced property management company.The key principle will co-sign the loan to help show the net worth and liquidity needed for the underwriting department.  

I have a cousin in Florida that is moving and would love to come to an agreed price on their place but just take over the mortgage instead of securing my own loan as I am looking for it as an investment and would be required to do 20% down (which I would not have). 

Assuming you can cover a decent portion of the $60K from the 401K loan, get some sort of loan for the rest.  

Further, how do I demonstrate proof of funds on a HELOC (ie, the equivalent of a pre-approval letter for a loan)?

Conventional loan How did you add value to the deal?

He couldn't sell the property because no bank would give a loan on it with that in the contract.2.