Yes. I don't recommend having a car while visiting cities. Next time limit your rental to visiting the countryside and smaller towns, which tend to have more limited public transportation options.
Unfortunately many North American travelers are not familiar with the International Road Signage system used in Europe, which is based on symbols rather than words, and often enter streets where only buses or cars with special permit (usually residents) are allowed.
The fine you got appears totally legitimate. The authorities, by law, have 365 days to notify a foreign resident starting from the time the rental company has communicated the name of the renter who was driving at the time the fine was issued. That limit is only 60 or 90 days (don't recall exactly) if the renter was a person residing in Italy.
If you don't pay there are 2 possible consequences:
1-If you enter Italy within 5 years (statute of limitations for traffic violations) and have an accident, or somehow you get pulled over, your name might come up and you may have to pay then a higher fine.
2-The collection agency you mentioned may be affiliated to an international network of collection agencies with offices in your State/Province. They will not take you to court, because it would be too costly to have the foreign judgement recognized by a court (that doesn't happen anyhow). However you will have to deal with the collection agencies' unorthodox (and often illegal) harassment practices. As you may know that industry is not well regulated and often those collection companies will make your life miserable in order to force you to pay. If they report you to a credit bureau you can probably have that removed, but it will be a hassle for a while.
If it's not a large fine, it might be better to pay, even if you don't plan to go back to Italy in the next 5 years. In terms of statistics, I can tell you that Italian authorities (and the collection agencies they hire) generally recover about 20% of the fines issued to foreign residents. Believe it or not, there are a lots of Italians who also don't pay them, and that number has increased exponentially during the recent economic recession.