The official currency of Croatia (Dubrovnik) is the Kuna. Croatia is not in the EU. The Euro is the official currency for the rest of your ports. I suspect the acceptance of the euro will be fairly widespread.
However, that said, provided you use small denominations, the greenback will be accepted anywhere.
I think there are 2 ports in Dubrovnik, do you know which one you will be using?
Rome city is approx 40 miles from the port. I never visited Rome by cruiseline - sorry I can't help there.
Livorno to Pisa/Florence will be easy by train. Livorno is 90 mins away from Florence. Pisa is often overlooked IMHO - the Field of Miracles has the wow factor, for me anyway.
MonteCarlo is so tiny - that is a walk in the park. However mot tourists for the cathedral where Grace Kelly is buried.
Barcelona - again do you which port you are using?
BTW you could always use your cc instead of currency notes. As I think of more, I will post back.