Not necessarily specific suggestions for you, but some comments.
I am not sure that the "star" system in the US means much from the stand point that there is no central authority that assigns stars and no consistent criteria, so your trust is not in the number of stars, but whoever is assigning them. Most cases though in Europe, there is a standard that a hotel of a number of stars needs to adhere to. However the number of stars usually speaks to the amenities rather than the quality of the beds, furnishings, or what the Hotel paid for the rating. It more likely means is there a bath in the room, elevators, food service, 24 hour desk, etc.; all depending on the system in use in the country that the hotel is in.
I have stayed in fantastic one and two star places...rated such because they did not have an elavator in a two story structure, no 24 hour desk, maybe a shower instead of a bath, and I have stayed in some horrendous 4 star places (well that goes for both Europe and the US) The only thing I can add is you need to know your own tolerance for comfort or expectations for a stay. It is different in Europe than in the US, I have many friends that would be shocked at using a shared bathroom down the hall, if the price is right, no problem here.
As for your situation, for your price range, having a balcony, especially overlooking something interesting may be a stretch, but mainly because most of the rooms just do not have them. If you focus on that, you may overlook many great places...but it is still worth considering when available.
As for destinations,
In Rome I usually stay at the Hotel Trastevere, simple place, but comfortable. Staying in the Trastevere is a bit out of the way, but a joy just to wander at night and a bit more of a neighborhood feel. Many others love the Campo di Fiori area or the Pantheon area, but quiet may be more important than balcony there.
Florence and Venice I have never really settled on a location or Hotel. If I were to have any advice, it would be to splurge on a place in Venice, maybe even double your budget, and get your balcony there, with a canal view.
In the Cinque Terre, any traditional measure of quality goes out the window. There are a few, more traditional hotels, but mostly you are looking at rooms, but more rustic, fewer amenities. A few of Ricks picks in Vernazza are a place to start, a few places in Monterosso offer the more traditional hotel experience, but I would push you to risk a bit and find an odd room in Manarola or Riomaggiore. A balcony is a bit more of an option there, maybe more worth the money.