re: Rome -insight on which neighborhood to stay

I am the definition of a novice traveler so plenty here can give you loads better advice than myself I imagine. I went to Rome about a year ago and my wife and I stayed in an air bnb near the fountain hosted by a lady named sylvia, see her page here: LINK

It's extremely close to the trevi fountain and we also walked to the coliseum and vatican. Didn't hit the others you mentioned. Sylvia was super helpful (spoke fine english) and even arranged a taxi to pick us up from the airport on our arrival and arranged one for us back to the airport afterward. Despite being super close to the fountain we didn't hear any of the hubub, it was perfectly nice and quiet in her building.

That's truly my only experience wiht rome so it's all I can provide, btu I had no issues whatsoever staying with her and thought it was a great value for the price.