Definitely think this would be too much. I know 21 days sounds like a lot but easy to spend a lot of time in many of these places. Barcelona and Rome should have a minimum of 4-5 days each.
I realize from your list you are probably planning on traveling from Barcelona to southern France then on to Genoa and Milan, but I think that whole part of the trip would probably be what I would cut out. I would try to just get a cheap flight from Barcelona to Rome or Florence and skip that other part - I think it will not be as relaxing as you think.

I love Barcelona - my second favorite European city after Rome, but I think if you really want to do the Italy portion justice you could easily consider spending the whole three weeks in Italy, particularly if you want the time to mosey around - which is what Italy is fabulous for!!

That is what we assumed. We had a couple tell us to fly into Venice, spend 1-2 nights there max. then rail it to Tuscany. We have heard that the Tuscan region, Rome, and Amalfi Coast were the most amazing and to focus on spending the majority of time there.

If you start in Venice, 3 nights and and then head to Florence by rail and spend two nights there, you'll have used up 5 nights of your trip. This is where I'd rent my car, (from somewhere just outside the City) and "mosey" through Tuscany for maybe 4 nights, by pass Rome and head for the Amalfi coast for 3 nights. That's about 12 nights. Drive back up to Pompeii on the way to Naples and two nights in Naples. Turn in your car here.

Catch the train to Rome and have your last week in the City. Believe me, there's plenty to see in a week.

Great recommendation Rubicund. Thank you that sounds wonderful! Do you have any must-see, must-eat, must-do recommendations for the trip?

There are many must sees in Rome. I suggest you get a guidebook (I always like the DK Top 10) to get a quick overview of the top things to see. I would also suggest you read people's trip reports here to see what they did and enjoyed. I have two trip reports from Rome with lots of details about what we did, where we ate and pictures too. You can read them here:

http://www.fromhometoroam.com/2012/05/rome-2005/
http://www.fromhometoroam.com/2012/06/rome/

The same is true of Venice, Florence and Naples, so splurge on guide books! On the AC, I'd stay in Positano-it's criticized for being a bit commercial, but it's the most spectacular of the AC villages.

Eating and staying depends on your budget and the above cities & Positano have a choice of hotels across the board. Try and stay nearer to the bus terminal in Positano rather than further up the hill if you like to be in the thick of it but further out if you want to be quieter.