I am so jealous, we were in Italy last spring break and it was one of the best vacations of my life! We rented an apartment near Piazza Navona. I promise you the BEST gelato is at Gelateria del Teatro.
We had gelato every day for 10 days (both Florence and Rome) and this was our favorite in Rome. March isn’t too busy so we never had lines but I have heard they can get lines out the door in the summer. Best part is that it is on a beautiful little shopping street and is walkable from Piazza Navona.
The only restaurant I can remember off the top of my head was called Nanas - it was close to the Trevi Fountain. We found it by accident, we were there at an off time and many restaurants weren’t open yet. Nana was open and had a big table outside - we loved it! Also loved that there is one of those fresh water pipes right in front of the restaurant. We didn’t know what it was, and in fact I thought it was kind of gross at first because it was just pouring water all over the ground right in front, then we saw people stopping to fill up their bottles! We had never heard of these fresh water spouts but they are all over the city. My son loved finding them and brought an empty water bottle everywhere so he could fill them up.
My suggestion would be to splurge if possible and get a private tour at least for the Vatican. It is crowded and a little confusing no matter when you go. The guide gets you through the lines and you get to see so much more than you would on your own. We saw tons of the group tours but it looked hard to keep up and to hear what the guides were saying even if they had the headsets. We also used the guide for the Colosseum and then he drove us to the catacombs and the Apian Way. We may not have needed it at the Colosseum but we did get so much information from him and we wanted to have a van to drive us out to the catacombs.
The tour guides we used were called Italy Chauffeur Service. We sued them for a lot of things (driving from airport to Florence, wine tour, Rome tours) some just driving and some with a tour guide. All of the drivers were great, not of our guides were absolutely excellent. Our guide in Rome was named Marco. He is an archeologist who only gives tours when he is home from his digs/studies - he was great.
If I think of any more restaurants I will let you know.