You could offer some sweat equity in exchange for on-the-job experience.

Again this is also going to be your primary home so maybe you are willing to concede a bit of your potential returns in exchange for the "I just love it though" factor but from a strictly investment perspective I would look for something more along the lines of a boring brick flat roofed building.And lastly:  welcome to BP!!  

Some investors came from a 1031 exchange on a 77 unit apartment complex we sold in the Mid-West.

I'm not sure where you heard that but you cannot acquire anyone's property by simply paying off someone's (delinquent) taxes.The only way theoretically you could do that is working directly with the homeowner and getting them to deed you the property in exchange for the tax payoff.  

I will work for free in exchange for knowledge.

Me and landlord have been exchanging emails they eventually come down to 25% of the cost which is $400.

@Dave Foster Is there an additional issue trying to 1031 a contract assignment now that the TCJA limits like kind exchanges to real estate only?

My partners and I went full cycle on a 27-unit in Phoenix, exchanged into a 92-unit in San Antonio.

What's my minimum amount I must put into 1031 Exchange(s) to minimize my tax burden?

They might be willing to release in exchange for you lowering/dropping your demand on the first one.