We evicted a tenant since they didn't pay rent for 2.5 months. 

If you rent it for 2 years and then sell it, you shouldn't need to pay taxes on the capital gains.

I am planning to get into the TN real estate market as my first investment and am looking for some information.In particular: Areas to steer clear of, property management fees, typical cosmetic rehab costs, etc.My strategy is to put large down payments on 2 properties (relative to housing prices) to help pay for my living in NY while still gaining appreciation in a growing real estate market.

A HELOC is certainly an option if you could pay it off rather quickly or else you'll end up paying a whole lot in interest layer one. 

When paying them , in our contract should we put that we will be paying them out of “rental value “ , “rent due” or “rent collected”

It may even take a 3-5 years to get cash flow positive but if during that time it is cheaper to own monthly, than pay rent, you are way ahead.  

This would limit your options for the financing packages you might be able to get or which lenders would be willing to work with you since most lenders will want to see 2 years of pay to qualify you.

For example, I have a tenant portal where they can pay rent or submit maintenance requests.

What is your price point and are you looking for turn-key properties or something along the lines of a "fixer upper"?

Another option is to invest in the market instead of paying down a mortgage.