The reality is you can rent more house than you can qualify on a mortgage.  

I spent a lot on improvements on a rental this year (hvac system, new flooring) that usually I would have captured on my tax return but I’m worried that the lower AGI is going to affect how much I qualify for for the new house.

You can also try to qualify for SNAP through Family Services.

I have started my own form of syndication (not formal, just investing with family), and wanted to remove a blight from the neighborhood while at the same time building momentum for my new business partners.

Is the bank giving back the difference to the owner or the owner needs to submit a formal claim?

One challenge you might run into is that a lot of lenders will want to see two years of rental receipts to count that towards your income so you want to make sure you will qualify for both loans.  

I'm still learning here, is this something that I qualify to look into, and is this a good idea?

If you dont rent out that 3rd unit and will stay in that unit till you purchase the new property then you will need sufficient income to qualify for the full mortgage (PITIA) of the new property with the net rental calculation of the current triplex using only 2 units (since youre living in unit 3) which may result in a much less favorable rental calculation than you had previously planned for.

Unmarried couples, and other "groups" we require everyone to qualify on their own.

Who would be interested in purchasing pre-qualified and 'willing to sell' seller leads in Nashville?