When is Passport or Driver's necessary when showing ID in Italy?

Driver's license has always done the job for me. You do need to show your passport when you check in to a hotel.

I've never understood what use it is to have a photocopy of one's passport.

Other than flying into a country when is passport required?

Most of the time at the hotel but Rarely in other instances; and I go to a lot of out door ticketed festivals abroad where security is tight as measure for all the crazy stuff going on in the world.

Also for "document" ID when visiting sites that I have to pick up tickets I ordered on line, can I just show my US driver's license?

Probably but that has never happened as the tickets in all my trips come print at home in the form of a PDF. Or you present the email and they scan the QR or bar code and then the ticket comes out
Make it so that you do not have to pick up tickets that you order online. And when you buy something read what it says that you will need to retrieve a ticket. sometimes it is just the credit card you used
My last trip I bought a beer on the plane and they asked to see an id with my signature and I just used my state id

And I agree with the above a photocopy of a passport is useless you can scan it and leave in your email and pull it up if the embassy asks for it to help you replace the one you lost; but I have never heard of someone who benefited from having a photocopy in any kind of situation traveling abroad

The only legally recognised ID is your passport, so you need to show this whenever you legally need to show ID: Entering or leaving a country, boarding an aircraft, checking in to an hotel, or (hopefully this won't happen) if asked for ID by the police. A copy will not suffice in this case.

Private individuals and companies can choose to accept (or not accept) whatever they want. For tickets they may want to see the credit card which was used to buy the ticket. Sometimes when you borrow things (headphones at museum is phone case) they want to keep something as security. They may (or may not) accept a foreign driving licence.

Times in Europe I have needed my passport other than at the airport:

Checking into a hotel. Italian hotels need to register each guest. In the "old days" they kept the passport for a few hours to do this; now, they just photocopy it and hand it back to you. In many other countries you have to fill out a form with the information (sometimes including passport number), but they don't have to actually see the passport (so, if you know the number, you're set)

Buying a SIM card. To foil predators, terrorism, etc, phone stores and internet cafes in Italy have to get a copy of your passport. Every country has different rules on this; in some you can buy a SIM card like a bottle of water, but others need a passport.

Not in Italy, but one time I was almost caught out was the Jewish Museum in Munich. I had to show my passport to get in, and didn't know this beforehand. I always carry mine with me, so I was fine, but I know many people don't. I've never had to show it at other sights - even other Jewish ones with extra security.

Not in Italy, but in France my train ticket printout specified that it had to be used with my passport (to prevent me printing out multiple copies and giving them to others).

Not in Italy, but my Swiss Travel Pass is supposed to have required a passport to be used. I was never asked for my passport with it, but other posters here have been, and I had it ready in case.

Whether or not to carry a passport with you, vs. leaving it in the hotel, is a debate that never ends. I'd estimate that responses on this board, whenever the subject comes up (about 1-2 times per week), are evenly split.

As for picking up tickets, each kind of ticket will specify what you need to pick it up. It may be passport, credit card, print out of the e-mail they sent you, a code number, or something else. Follow the exact directions for the ticket you purchased, as the needs are different for each ticket. I've never bought an online ticket for something in Europe that could be picked up with a driver's license (I suppose it's possible).

Just to walk around town, or to go to a beach nearby, I leave the passport in the hotel safe. Carry a photocopy with you just in case. While traveling from city to city, it may be better to have it with you. For example if while in Rome, you want to take a day trip to Florence by high speed train, it might be a good idea to have your passport with you rather than 185 miles away at your hotel in Rome.

Besides when mentioned by others above, you need it also to drive a rental car, in case you get pulled over.