Have to agree that this is the most insane itinerary I've seen in a long time as well. You'll barely have time to get your bearing before moving on, and will be spending a lot of time in stations and on trains, and running around finding places to stay the night, not to mention that the most attractively priced/rated in the most convenient locations will probably be long booked up.
Additionally, there are many attractions you can't just "pop into" during high season in Europe. Ticket lines for the best of them will be very long so advance tickets are necessary to avoid wasting a lot of time standing around. As well, there are days main attractions are closed so if you only have 1 day and land into the city on that day...
Possible snags with train schedules have already been addressed. Add the possibly of a strike or two to that picture. Overnight trains - even those that are still operating - are of no use to you in Italy as your chosen cities aren't far enough apart to need them.
I see potential issues right off the bat with Capri; if you are arriving in Naples that same day, be aware that it's going to take considerable time to get there, and it is a very expensive place to spend a night. Boats back to Naples leave relatively early (currently a little after 8:00 PM but summer schedules aren't up) and will be jam-packed. I'd skip it and add that day to Naples. Heck, I'd drop Naples/Capri altogether, fly directly to Rome and add at least 1 day to that city; lots to see there.
The Cinque Terre will be mobbed and accommodations in the 5 villages long booked up. You could try finding a bed in La Spezia or Levanto but personally I'd skip it or add one of the dropped Capri/Naples days. If you must do it, your itinerary should be Rome>CT>Milan. It doesn't make sense to go that far north only to head south and then north again to Vienna.
Another alternative would be to drop Naples, Capri and the CT, add a day to Rome and the other two to Florence, which is not on your list but well worth doing, and will be quicker to get to via fast trains from Rome. Take a fast train from there to Milan.
Drop Copenhagen; add that day to Prague thus eliminating all one-nighters but Milan. If you leave Florence early, you can cover just enough of the highlights in a partial day.