East Coast of Italy

I'm confused. You're renting a car but you don't want to drive? Do you just mean you don't want to drive from Rome to the Adriatic Coast? If that's your question, of course you can take a train to a city on the Adriatic Coast and pick up a rental car there. Why wouldn't you be able to? High marks on this website for autoeurope.com for car rental. (Do NOT confuse them with autoeuropa.com, a separate company that gets terrible reviews.)

Yes, you can fly into Rome and take a train to the East Coast of Italy. We have family in the Rimini area and have done it a number of times. We used to fly into Milan because it's cheaper to fly into there, but Rimini is pretty far North. You can arrive in Rome, take the train to the largest city close to your destination and rent a car there. However, the Rome airport is far from the actual city of Rome, and it may be the least expensive place to pick up and drop off the car, and avoid driving in Rome just take the major highways to your destination. Not sure what your actual question is if you're staying in Rome, and want to rent a car from there, you can certainly take the Leonardo Express to the Rome Airport and rent there outside of the city.

Depending on where on the east coast of Italy you want to be, you may be able to fly there with a plane change. Pescara, Ancona, Bari, and Brindisi have airports, for starters.