Hi Meghan,
Congrats on planning your upcoming trip to Rome! Great questions about vegetarian and celiac-friendly dining choices. As a celiac, I too have been digging for recommendations and resources on gluten-free restaurant choices for my upcoming trip to Rome, Florence, & Prato.
The Puntarella Rosa link kindly provided by Roberto is a great resource for planning. Check out this link as well:
https://www.scattidigusto.it/2016/05/11/roma-migliori-ristoranti-celiachia-gluten-free/. Both links were recommended to me by someone living in Rome.
If you find a place with gluten-free dining options in the Monti area, definitely let me know. The closest I have found is Taverna Barberina right next to Palazzo Barberini. Close to Monti, but more so the Quirinale area.
For the Colosseum area, the closest place I have found is http://www.centopercento.bio/. Ran across that one last week, and it would check all the blocks you're looking for with vegetarian and GF options.
I hope this helps and happy planning!