Italy has very stiff drunk driving laws, so whoever is driving the car around Chianti can't drink wine. Also in Italy the driver needs an International Drivers Permit accompanying his/her license. The IDP is easy and cheap at your local AAA/CAA office, about $25 including photo.

As bobthenavigator says, you can't just drop in at a winery in Chianti and taste the wine. Most if not all aren't set up for walk-in visitors. You must make an appointment first. As Bob said, a better bet are the enotecas in towns, which offer a variety of wines.

Re: the ZTLs, the problem will be driving out of Florence and back in. If you get just a standard rental car, as I suspect, you'll be subject to steep fines if you happen into a ZTL, all too easy in central Florence.

I would say skip the side trip to Lucca and just enjoy all that Florence has to offer. There's so many things to do there and one day of exploring Florence and its culture is not enough so I would just relax there my last day.