I don't know much about the drive from Amsterdam, but I just got back from Rome last night. We spent 10 days there. Driving in Rome is scary. The roads are confusing, and there is always traffic. Not only do you have to deal with cars, but there are tons of mopeds--they have a death wish. If you are only dropping the car off, you should be fine. But good luck with directions--bring a good map and an English-Italian dictionary.
The only gotchas are tipping at restaurants. In most of Europe, service and a cover charge are already included in the bill. It is customary to leave a small amount of change on the table as an extra. In Rome, they love to tell you that tip isn't included, but it is--all over! So just leave about a Euro or less on the table. Try to see all the major sights, but also go to Nero's Golden House--Domus Aurena--it is great. Have you ever thought about taking a train, instead of driving? Have a great time!!!

rent a fast car for the Autobahn

Thanks for the advice Jodi.

Actually we are taking a train from Amsterdam to Heidelberg Germany. And driving from there to Munich and Garmisch then to Rome. We thought about the train to Rome but wanted to see the "sites" on our schedule during the drive down. Might end up being a major mistake but I figure we can find our way in 12 days