Italy with family including a 13 year old ad a 7 year old

Lucca is a fine place to stay without a car. It's a little over an hour by train to Florence, so if it's Florence you are more interested in, I'd lean toward staying there. If you just want to pop over one of the days and have a more relaxing trip, then Lucca.
Are you flying in and out of Pisa, heading to Sicily from there? Many will also make the argument to put the time in Sicily since you are going there anyway and it is such a large island with so many places to see. But if you would prefer to break up the trip on the mainland, no problem!
Lucca is so small and walk-able, I'd prefer the simplicity of staying within and not having to rent a car for such a short stay, but if you would prefer a farm stay, there is parking around Lucca (be sure you know about ZTL). For Florence day trip, I would still use the train.

Wow. So fast! Thank you. Let me explain a bit more.

The plan at the moment is to fly to Paris, then fly to Palermo for the wedding, spending 4 days in Sicily. Then fly to Lucca and spend another 4 days exploring with a day in Florence, a day at Pisa, perhaps a beach stay, perhaps a cooking class in Tuscany and then fly to Paris for another 4 days before flying back home.

Yes with your advice I think we won't bother with a car, use trains, and stay within the city.

Does this sound do-able?

It is honestly something that I would not do for myself--I don't like flying on vacation beyond getting to and from somewhere, then I stick to land or sea travel within a smaller area. But, I do realize that flying to Sicily is very expensive compared to flying to Paris, and you might think you won't be back for a while and want the variety.
I would consider sticking to Paris and Sicily, as the kids might have more jetlag, etc. But, if the flights work out, it is not unreasonable, just giving my opinion.

Thank you. I will share with the family. I think the idea is to try to minimize travel but give a taste of a few places.

Those are not mutually exclusive--where is the wedding? Sicily fans can give you all kinds of options for adding variety without adding trains and planes. But your group will know what works best for the family, good luck!

Why not stay longer in beautiful Sicily? Ortigia Is off of Siracusa is wonderful. We spent a wonderful week there.
We have stayed in Lucca and it is lovely. However, it is so far west in Tuscany that we didn’t like it as a base. We stayed put in Lucca. Look at Siena for a better location in Tuscany to explore the region although you don’t have much time.

how many is the group? I see you mentioning at least 3 generations. How many is that? Thinking of travel issues.

There are 6 of us, yes, 3 generations. Girls ages 7 and 13, their parents and us, grandparents who are all robust travelers and in good health.