Then, the developers offered to sell our equity group the deal out-right in exchange for settling the debts and payables on the project and returning current investment principal, including a negotiated return on said principle (between 50%-100% on the money already in). 

This could complicate or delay the exchange, especially if dealing with a bank. 

Refi's though you have to qualify and have provable income.If you hold for 2+ years as an investment you can do what is called a 1031 Exchange and buy another like kind property and transfer the tax due.

I think you can use funds to pay some of your closing costs but I've heard conflicting info on this.I did a very similar exchange last year and I would caution to keep an eye on your reserves.

Someone like this may give you a small equity stake in a property in exchange for handling all of the operational duties.

@Dave Foster will be along shortly and explain a 1031 exchange. 

Need help, this is our first exchange

Third, just for reference, in case you did inherit a rental years ago, the name on the property purchased with a 1031 exchange has to match the name on the selling property.

My question is - can we do a 1031 exchange to use the equity in the condo for the purchase of the new home so that he and his mom can avoid the income taxes, and then he quitclaims the title of the new home to his mom?

In the future, we plan to build a secondary exchange on which these investents can trade such that investors can sell their stake before the property is sold.