You don't say what your interests are, but based on what you've already seen / done, I'd add a visit to at least one of the historic homes / properties in the area. Belle Mead (with the winery), Cheekwood (with it's fabulous botanical gardens), The Hermitage (home to President Andrew Jackson), Carter House / Carnton / Lotz House / Travelers Rest (all Civil War sites in or towards Franklin), Belmont Mansion (on Belmont University's campus), Croft House at Grassmere (included with admission to the Nashville Zoo - a great destination unto itself). Also a visit to the State Capitol / Bicentennial Mall / State Museum Military Museum (The main Tennessee State Museum is closed as they ready for a move to their new location near Bicentennial Mall in mid October). Near the Bicentennial Mall is Farmers Market, which will check off the food part of your request as there are multiple unique ethnic eateries there in its food court. Centennial Park with the full scale replica of the Athenian Parthenon. A visit to Radnor Lake State Natural Area will make you forget you're in the city limits of Nashville.
As for other food places, you hit most of the iconic tourist places. I'd recommend one of our meat and three diners (Arnolds and Dandgures near downtown are two great ones. Also, one of our classic burger diners, Rotier's or Brown's are my two favorites. They're simple, flat-top cooked burgers with only traditional condiments, but deliciously simple.
I agree 100% with Arnold's and with either Rotier's or Browns. Get the roast beef with mac and cheese and greens and banana pudding for dessert at Arnold's. At Rotier's and Browns, just get the cheeseburger with fries. It is perfect at both places and choosing a favorite is like choosing a favorite among your kids. I would add Monnells to the list, particularly for breakfast. It is family style at a large table, so you never know who you will dine with, but that is part of the charm. The food is ridiculously good. Fried chicken, sausage, bacon, country ham, cheese grits, eggs, potatoes, pancakes, biscuits, gravy and probably more I can't remember. Martin's BBQ is also a great choice - get the redneck tacos.