I'll respnod to the things I know about ;)
Rome - we stayed in the Trastevere neighborhood and found it ideal. Its popular in the evenings but not as much during the day as places across the river around Piazza Navona, Pantheon, etc. BUT, we found it ideally situated equidistant from almost all the sites (except Borghese museum/park*). We walked from our apartment in Trastevere to the Vatican (20-30 minutes - we had a 9 year old with us so pace adjusted), we walked to the Coliseum (again 20-30 minutes, and we did actually take the bus there on the way and it took far longer than walking back). We visited the Capitoline museum the following day and walked there and back, no more bus), the Pantheon, Trevi fountain, Spanish steps, etc. I thought the Trastevere neighborhood was like the hub of a wheel, and al the sites were around the rim of the wheel.
*the day we visited the Borghese we took the bus there, then spent the day in the park, and then walked all the way back to our apartment, passing the Spanish steps, Trevi fountain, Pantheon, etc (many we'd already seen). Point is, the walk from the Borghese to Trastevere is totally doable. We ended up thinking public transport was more laborious and time consuming than it was worth - and we had three kids doing that walking
Florence - we stayed in an apartment here as well. It was two blocks from the river, a few blocks from the train station in the direction of the Duomo, basically, it was great. I don't have any info. on hotels
Cinque Terre - I personally would not do this as a day trip from Florence. The train ride is 90 minutes. The appeal of the Cinque Terre is the natural beauty with the beauty of the villages, so you will want to hike and explore the area. I would stay at least one night, probably two. We hiked between Monterosso (where we based) to Cornelia the first day, stopping in Vernazza for lunch and swimming mid-day, and trained back. Then trained to Riomagg. the second day and hiked up and over the pass (the coastal trail was closed and probably still is) north to Manarola for lunch and swimming the second day and trained back to Manterosso. But look at a map of the area and descriptions. You can just train from village to village and see them, but to me thats a tourist-packed, hot cruise-ship type way to experience the place (yes, I don't like cruise ships).
Speaking of cruise ships, you can google to find out when the cruise ships dock at the CT (I can't remember the title of the search) and visit on days when there aren't cruise ships! Basically, there is a website that tells you the cruise ship docking schedule for the whole mediterranean ...
best, jessica
edited to add: this is me responding NOT to your whole itinerary. I didn't pay attention to how much you have in mind....
