Italy in 7-8 days

Is that 7-8 days or nights? Best to count nights with the distances you plan to travel. You listed five stops in 7-8 days, with a young child. I would just stay in France and save Italy​ for another trip.

If you must go to Italy on this trip, I would suggest Venezia and Tuscany.

You listed five stops in 7-8 days, with a young child.

Yes, I'm afraid that's MUCH too much for the number days that you have. You'd spend most of your time just moving from one location to the next and seeing very little at all! And yes, count number of nights versus days. Traveling with a very young child also needs to figure into the equation as it involves added flexibility to work around their needs.

Pick two, such as:
3 nights/2.5 days in Venice
5 nights/4.5 days in Florence + 2 day trips via bus or train (Sienna and Pisa/Lucca; not the Cinque Terre.)

Or....

4 nights/3.5 days in Florence + 1 or 2 day trips
4 nights/3.5 days Rome

Travel between any of these should be done by train.

The first thing that comes to my mind is how are you going to sightsee with a toddler? Stroller or carrier? Rome and Florence are fine with a stroller, fairly level and good paving for the most part. Venice is more of a challenge since you'll be going over bridges all the time. Rome and Venice are hot and humid, Florence is hot, don't know if it's humid. Even with 8 nights, I would stick with 2 cities. You can day trip from Florence to other Tuscan towns and both Rome and Florence have lots to see and do.

Agree with Chani, have done Venice with my then 3 yr old granddaughter. The bridges are a killer and there were 4 of us.

First, fly multi city, into Paris out of Rome.

Use vueling air to fly between Paris Orly (easier to get to than CDG) to Florence. Just be sure to check their luggage requirements.

Many good day trips from Florence. By bus, Greve and Siena. Around an hour to each. By train, Bologna, (food capital) Pistoia ( cute midieval center) Montecatini Terme (spa town, beautiful fountains) these two could be combined.
And Lucca, (walled city with ramparts). Other than Lucca all under an hour. The only problem is choosing where to go with your limited time, since Florence has more than enough for your whole Italy excursion.

With Rome and a child, I would not even think of doing the Vatican Museum. Just wandering this city and touching on a few highlights will keep you busy.