awesome, rome is a cool city, make sure to keep ya colla’ popped

For days 3 through 5, you could probably just stay in one town on the Amalfi and take the train, if this helps at all. This is not to steer you away from Naples, though, especially since Naples has the museum with all the artifacts taken from Pompeii.

A.) Vatican City is in Rome, or technically surrounded by Rome.  You don’t need to consider it a separate destination from Rome.
 
B.) I like Cil’s idea of staying in Amalfi or a nearby town rather than Naples making both Naples and Pompeii/Herculaneum as day trips.
 
C.) I think you should stay overnight in Capri, since it will be quite the commute (bus or train to Sorrento, then ferry to Capri)
 
D.) can you extent your time in Italy by a day or two?

I’d actually keep the one night in Naples just for the experience. I visited the “Bombay of Europe” last month and loved the city. It ain’t pretty but it had bags of character.
 
Also the archaelogical museum is a must see to really add understanding to your Pompeii visit. 
 
Looking at your itinerary your schedule is very tight on day 3.
 
Perhaps consider travelling from Rome to Amalfi that day. Spend the afternoon and evening lazying in that stunning place.
 
Day 4 do your Capri day trip.
 
Day 5 travel from Amalfi to Naples calling in on Pompeii, Vesuvius and even Herculaneum if you’re not too tired.  (Personally I’d skip the Vesuvius ascent.)
Arriving in Naples. The Archaelogical Museum is open until 8pm so you could squeeze that in also! Pick one of those famous pizzerias and eat the best pizza you’ll ever likely to eat!
 
Day 6 travel back to Rome