Can my adult son to live in my 2nd home (has no mortgage) and pay "rent" to cover insurance and property taxes without claiming the "rent" as income? What can I deduct?

What you have is just family living in your second home and helping with expenses ... you do NOT have a rental so do NOT enter this on your tax return.

As to what you can deduct ... the same as you did before but with the new itemized deduction laws.   

See this TurboTax support FAQ for how tax reform will affect your 2018 tax return - https://ttlc.intuit.com/questions/4482394-how-will-tax-reform-affect-my-2018-federal-tax-return

And since you don't have a mortgage on the second home, your only deductible item will be the property taxes - subject to the new tax law's dollar limits.

Does Son plan to claim a State tax rent deduction/credit?

The property is in TN, no state tax, so no plans to claim this credit.