That itinerary will give you long, tiring days, but if that's the timing that works for you, go for it. I would strongly recommend using a private tour service for Rome, and maybe for Livorno as well. We were able to hit many highlights of Rome (Coliseum, Trevi fountain, Pantheon, Vatican, and a few more) in a single day with a private guide. Not having to wait for everyone in the group to be ready to move to the next stop is an advantage, and having a car with air conditioning could help with the heat.