We like to stay in an area loosely defined by The Pantheon and Piazza Navona. Two hotels meeting your requirements during your stay are Hotel Smeraldo and Hotel Ponte Sisto. Although both are listed on various hotel booking sites, I suggest you check their own websites for best pricing.
Rather than leave transfers up to your hotel, I suggest you book them yourself. There are other good ones, but we always use RomeCabs. Their pricing is usually better than what your hotel will offer and having direct contact with your provider in the event of a flight delay is a plus.
This past September our flight from EWR to FCO was delayed by three hours. I notified RomeCabs and they adjusted their schedule to accommodate our late arrival. I don't know how it would have worked out if I were working through a middleman, the hotel, for our airport transfer.