Depending on what time of year you're going, remember that Iceland Air goes to Europe via Keflavik & on fares to Europe, I believe you get a free stopover for upto a week in Iceland. Driving is on the right side of the road & you can camp almost anywhere if you want to do it superbudget.

I'd recommend flying into a major city (not London - they just significantly increased taxes to fly direct into the UK from the US) as fares will be cheaper. I'd recommend flying into Amsterdam as it's cheaper. From there, use the train to travel to city centers. Renting a car isn't necessary if you're trying to hit major city centers as flights & trains will do - but, if you want to have a great experience, rent a car & roadtrip. You said Italy is a must - when I was there with a buddy of mine, we rented a car for a week and drove from Rome through Tuscany & up to the Alps down into France, hitting Nice & Monaco, and then back down to Rome - we did that roadtrip in 6 days & I saw more of the Italian country-side & wine region than I would have had I not had a car.

Here's a link to some cool country-side hotel that aren't really supercheap, but they aren't as expensive as city center hotels - http://www.silencehotel.com/.

Depending on the fares, time of year for travel & your travel style, which is budget I'm assuming given you've posted here, I'd recommend doing US-Iceland (3 days) to Europe. In Europe, I'd fly into Amsterdam, spend 3 days there, then take the train to Paris, spend a few days there then down to Italy & see Rome & Venice & the Tuscany region - spending about a week there since that seems to be a focal point of your trip.