Ciao Paul,
Parking in Pistoia and then hopping on a train to get into Florence makes sense, makes it less of a hassle to just avoid traffic and finding parking in Florence (although there is a new parking lot just off the A11 highway as you arrive at the airport in Florence, then from there you hope on the T2 tramline that brings you to the center of Florence as well).
Not so sure you need to do that for Lucca since it's a smaller town, the historical center is enclosed by walls so there is no chance you can drive in without realizing it. Just drive there and park outside the historical center, walk in. The same for other smaller towns. As soon as you approach the towns, there are places to park and signs that will indicate parking lots, so if you generally head to those first, you never run the risk of driving into the off-limits driving areas that are active in the oldest parts of any town. We drive to Siena, there are lots of parking lots around the city and park at any of them, then walk in to the center.
As for Verona, if you were to take the train, you'd first have to go to Florence and switch trains. So you should also consider whether you could drive all the way.
As for leaving the car overnight, it is generally safe to do so, like all residents do. I'd just make sure you leave nothing inside the car that could attract attention to being broken into. Just make sure you check well where you park, whether it is a paid parking and cost for overnight or street parking, checking if you need to pay the meter for daytime hours as well as night cleaning schedule (most streets have a day of the month where street cleaning is done so parking is not allowed, risk getting it towed). Plus one night is okay, too many nights in the same spot without moving it could attract attention.
Let me know if you have any other questions!