We are wondering what the Monteverde Nuevo area of Rome is like. is it a good area to stay in? Does it feel welcoming and safe? How far is it from sights like Trevi Fountain and others? Is it walking distance to the center of things?
Suggest you use google maps ( or similar) and plug in the address of the lodging you are considering and looking at how far it is to the sites you want to see.
Is it walking distance to the center of things?
Depends on what is 'the center of things' for you.
I don't know anything about this area but glancing at a map, it's quite some distance from the things most tourists want to see. Walking distance? Depends on what sort of walkers you are but just eyeballing it (rough estimate) you're probably looking at 2-4 miles on foot from most main attractions, and that's "as the crow flies": it'll be more when winding through Rome's less-than-straight streets.
That "area" appears to cover about 1.5 miles from north to south so even where you're staying in it could make a difference in distance. Personally, it's too far out for me to want to hoof it daily to center and back - and we're stout walkers - so I'd count on needing to make frequent use of public transit.
For a tourist/traveler, this is a remote area. Sure it is within the city of Rome, it is residential but, depending on where exactly you are located, it will take 25 minutes by bus and an hour walking to reach Campo de' Fiori. It is farther to the Trevi, Colosseo, etc.
Find your proposed lodging on Google Maps then use the Directions function to plot routes to various sites and see how long it will take. It is, IMO, not a good location if you plan to return to your room during the day to rest (much advised!).
We lived on the other side of Rome and there was a cute little hotel with low prices nearby. We constantly encountered people staying there who had no idea how long it would take to get to the center and who were frustrated by the bus service as frequently a scheduled bus would not show up and we'd wait 20 or 30 minutes longer just to get the bus to take us 30 minutes to the center. At night, they'd be in a Roman residential neighborhood, which has its charms, but was not convenient for evening outings unless you used a taxi.
If this is your first trip to Rome, try to stay somewhere more central. Saving money on remote lodging will cost you in time.