Florence: Parking and Avoiding Evening Traffic, Going Southbound

I would likely "pick up" the rental car and stow luggage in it - make arrangements with rental agency to leave car there and take a taxi into Florence and return to car rental by taxi after a few hours in Florence. A taxi from airport to city center is €20 and your bags should be fine in the car at the car rental agency.

As far as traffic I can't answer that one but I am sure someone will chime in

How old are your kids? Unless they are very young, tell them to get a taxi and meet you at the hotel where you are staying. Then you can go and pick up your car later. Your hotel will hold your luggage after check out.
Traffic will be heavy within the city before 8pm. So if you take the Firenze Sud A1 entrance, there will be some traffic on the Lungarni to Via Marco Polo or to Viale Europa if you take the Verrazzano bridge. After that it will be nothing compared to what I have to face daily in Silicon Valley. Italians generally don't commute that far out of the city, and when they do, typically not by car. After 8pm there will be no traffic.
I would probably park in the underground parking on Piazza Beccaria or the one behind "La Nazione" Newspaper (enter from Viale Giovine Italia).

Thanks Roberto and Christi.
Roberto, is there no difference in traffic from a southbound freeway entrance and the airport, then? I like your suggestion of picking the car up later (kids are college students) if traffic is the same situation regardless. We would need to leave Santa Maria Novella for the airport car rental by about 4pm.

The airport car rental stays open until 11pm or Midnight, so you can go later, just arrange the pick up later. The traffic will be heaviest between 6p and 8p. But it will limited to the city streets, because the traffic won't be bad once you are on the freeway, unless of course there is an accident. It's 19:20 (7:20pm) now in Florence and checking traffic on Google Maps. I still see heavy traffic in the city but the A1 freeway is largely clear, except for southbound just before the Firenze Sud exit (the last freeway exit in Florence), which is often heavy at this time.