I agree that Venice and Tuscany are too much...with muppets :). It will be hot, hot hot! :) Okay here is my BOOK :)
In my opinion Cinque Terre with young children is too much walking and crazy crowded. Everything is up and down and hiking which is the point but not in heat and with tons of crowds and little ones.
I have two suggested trips
Fly into MILAN, if you are LAKE & MOUNTAIN Lovers consider this..
Fly into Milan head to Lake Como day one ( 3 to relax and skip Milan 2 if you want to see Milan 1/2 day before train to Florence)
Day 1 sleep Lake Como pick a town!
Day 2 sleep Lake Como ( add a day if you think the family will want to get over jet lag etc)
Day 3 Train from Lake Como to Milan switch for train to Florence fast train 2 hours ( see Milan..Duomo ) or not
Day 3, 4, 5 Florence proper try to stay somewhere with a pool or there are some places just on the outskirts with shuttle service that may be cooler with gardens...option to schlep into other towns but Florence has a lot to offer
Day 5 afternoon or morning if you are ready to move on Rome
Day 5, 6, 7, 8, 9 Rome
Getting hot and want a change...see archeological sites outside Rome stay Day 8, 9 Stay in Ostia for beach time before flying home..you can go directly to Rome airport from here easily.
This is 3 hotels 4 if you stay in Ostia...I think this makes sense with children but still gives a wide view of a beautiful country!
Not sure if I got ALL your nights but just suggesting the flow...if small town Tuscany is on the bill then I am sure you will have lot's of suggestions here, it is not my thing...I love cities, art, architecture, markets and beaches!
Second option: if you are BEACH lovers...
Fly into Naples and out of Rome/ Naples/Islands with beaches and sites, Rome
Arrive Naples
Sleep NAPLES one night choose hotel near train station ( I like Una Hotel Napoli very safe, roof top view with breakfast included and soundproofed) Have pizza with kids, see Archeological museum ( feeling spry go to Herculaneum 25 minute train ride 15 minute walk from station to site, visit 1-2 hours, there is shade, the kids will be able to see tiles and urns and wells lots of easy walking compared to Pompeii. You could skip this altogether as Rome has tons of archeological sites and see these on a future trip.
If you won't TOUR Naples than stay in a hotel with a pool near the port area and take a very early ferry to the island of your choice Ischia or Procida...both have real sandy beaches that you just WALK on to, no cliffs or hiking to reach them!
I recommend ISCHIA for families Castle Aragonese is fun for kids and you can do 1/2 day at beach...stay in Ischia Ponte or Porto area with kids it is just easier on a short trip there is a boat trip around the island that takes you to all 4 major towns...recommended..or PROCIDA smaller island walkable but fewer sights...
Night 2 Island
Night 3 Island
Night 4 Island ( you could do a boat trip around Ischia go to 4 towns or day trip to Capri from either island)
Morning of day 5 take ferry back to Naples figure 1 hr and FAST train to Rome1.5 hrs ( this is when you COULD lock luggage at Naples Central station and see archeological sites, and museum pizza etc if better jet lag & kiddo wise)
FAST 1.5 hour train to Rome/every one is refreshed from relaxing and on the correct time zone to tour ROME!
Night 5 Rome
Night 6 Rome
Night 7 Rome
Night 8 Rome
In my opinion the kids will love Rome...Vatican City needs a 1/2 day at least, if they can walk more then return a 2nd day...
There are some affordable hotels in Rome with pools...
3 hotels ( 4 if you stay one night to get it together in Naples if not plane taxi ferry to island on first day is a lot...I have done it without kids many times but it kicks you!)
I have done these trips many times with the exception of Como...I usually go to south of France from Milan but that is another story!