1) 90 % of Airbnb apartments, especially in the center, are managed by companies. Therefore you will never meet anyone, maybe just a boy for the check-in. Noone to ask indications and get recommendations. Look for properties that are managed by the real owner. You will get a much better satisfaction
2) Given the lenght of your stay, I suppose you want to understand the real Rome; and you will have time. The areas you selected are no more the real Rome because they are flooded by tourists.
3) In my opinion, and I am Roman, most of the things to see and do are in less crowded areas of the center of even out of the center. E.g. did you know that rome is the European city with the longest coastline and that part of that is a natural reserve?
4) Given your budget, you will find excellent opportunities just out of the Aurelian Walls at no more than 100-150 per night. You will have enough money to use a cab all the day and see much more, and, if the driverr speaks English, to get some good piece of information.
5) Out of the area that you have indicated you will find better and chaper restaurants, cafè,shops.

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