For Paris, take a look at this pass. It will save you both time and money.
http://en.parismuseumpass.com/rub-t-price-36.htm
Rome is a little tricky, since you need to know the cost of each site and you only get 2 free with up to 50% on other sites. But it does include city busses and metro. The key is to use the most expensive sites first.
http://www.romapass.it/en/FAQ/
List of fullprice & discounted sites.
http://www.romapass.it/files/doc/SitiAderentiCostoBiglietti201809_eng.pdf
Florence has a pass too, the Firenze Card but At €85/72 hrs, it's hard to get enough sites in for it to make it pay for itself, but the skip the line feature may make it worth it for you.
http://www.firenzecard.it/index.php?lang=en
As for your itinerary, add as many days as possible.
This path works well transportation wise as well as alternating large cities with quieter scenic areas.
Arrive Rome. With a 13-16 hour flight, you're going to be cranky, tired or both. Check in, wander a bit, have dinner and go to bed. This day is basically wasted and that may carry over to the next day.
5 Nites Rome.
A good day trip is Orvieto.
Day 6. Train to Cinque Terre. 3hr30m.
3 nites Ct.
Day 9. Train to Pisa 1hr38m. Drop luggage at station, bus to Piazza Dei Miracole. Bus back to station, pickup luggage, train to Firenze SMN 1hr.
4 nites Florence.
Good day trips, Siena, Bologna, Pistoia.
Day 12. Train to Varenna. 3 hr42m.
3 nites Varenna.
Day 15. Train to Milan Malapensa. 2 hr.
Flight to Paris Orly, check Easyjet and Alitalia.
1hr25m.
4-5 nites Paris.
Versailles or Giverny if you need an escape.
If you prefer the Dolomites to Varenna. Use this version from Florence.
Train Florence-Bolzano. 3hr11m. From here it's a bus ride to Ortisei.
https://www.val-gardena.net/en/questions-and-answers.asp?id=244
3 nites Ortisei
Train evening. Bolzano-Venezia 2hr39m
1 nite Venice or at an airport hotel.
Fly VCE-Paris Orly. 1 hr45m.