I’d agree with the feasabilty and method of transport as described above, but I’d put in a plug for Piazza Navona & the Pantheon, or the Roman Forum as worthy drop in spots. But keep an eye on the clock: with international connections, security checkpoints, and public transport, 10 hours will go fast. It’s enough time for a few hours to grab a gelato, panini, espresso, have a look around, and then back to the airport.
I’d call it a library if it is a room for books. I had lots of books, and even lots of shelves, but I didn’t have a library until they came together in one room, with a place to sit and read them.