Maybe you can do something like flying into Paris. Spend 4-5 nights there, then fly to Venice (or somewhere towards the north of Italy) and work your way down to Rome. I wouldn't bother with Milan, and if you want to do CT then you'll have to skip Venice and/or Florence. Figure maybe 3 destinations in Italy at most. Forget Nice for now.

Use JanisJ's formula for your itinerary. That's a reasonable pace. The point from Adrienne about moving around less to save money is an important one.

to think you can get see so many places in such a short amount of time is a common "mistake" 1st timers make (that i often see posted on this forum). So I say this gently... The logistics of moving city to city, pack unpack, local public transportation, find hostels, etc. it all takes time. You can't say "2 days Rome, 2 days Venice, 2 days Paris" and have it make any sense.

It's hard to see the countryside and visit small towns by public transportation. I would suggest to streamline this first trip you skip that idea, except for Cinque Terra if you're determined, and just use the train to get around. Plan your itinerary around easily useable trains.

Did you have your flights backwards? You said you'd fly into Paris and out of Rome, but your proposed itinerary seems to be the opposite, starting in Italy and ending in Paris. I would cut out Milan and Nice and spread your trip out among the remaining destinations.