Do hotels in Italy (Rome in particular) generally let you leave your bags for a few hours after checking out so you can do more sightseeing? I know you can leave them before check-in time. I'm planning our train travel now and this occured to me. Thanks!
Yes, do it everytime.
Same. Never had a problem. You might like to check with the individual hotel to be sure.
Another option is to take your bags to the train station and leave them at the left luggage area. This saves a lot of time at the end of your sightseeing day if you are going to end up closer to the train station than your hotel. They do charge for this service, but sometimes it's easier than leaving it at the hotel. Also, ask the hotel where they store the bags. Our hotel in Venice ended up putting them in the lobby because they ran out of room in their closet. Luckily they were still there when we returned.