Do you have 8 nights ON the ground in Italy or are you including your overnight flight in this count?
If not, I see a schedule like this:
Land in Rome
Night 1: Rome
night 2: Rome
Night 3: Rome
Night 4: Amalfi Coast
Night 5: Amalfi. Coast
Night 6:. Amalfi Coast
night 7: Amalfi Coast
Night 8: Rome
Fly out of Rome
You don't want to be on the Amalfi the day of a flight from Rome, IMHO. The more efficient alternative to this would be head immediately to the Amalfi upon arrival and end your trip in Rome to avoid a single night in a pre-departure hotel. It will be a long day but worth it, I think, to eliminate a move.
Land in Rome, trains/buses/transfers to the Amalfi Coast
Night 1: Amalfi Coast
night 2: Amalfi Coast
Night 3: Amalfi Coast
Night 4: Amalfi coast
Night 5: Rome
Night 6:. Rome
night 7: Rome
Night 8: Rome
No, I would not squeeze in another city
No, I definitely would not rent a car. You do NOT want one for Rome, and driving/parking the Amalfi during high season is not my idea of fun.
Research hotels in www.booking.com (my personal favorite accommodation booking site) although I'm sure others will have different suggestions. You will need to decide where you want to stay on the coast? Everyone has a different favorite but Sorrento or Salerno are best for varied transport options as there's train service to both.