I think that my husband and I would find that itinerary too hectic, with too much time spent in travel. It's not just the flight time, but also time to get to the airport, go through security, wait for your flight, deplane, get from the airport to where you're staying, et cetera. If I only had ten days, I would go to one destination and just enjoy it thoroughly. Or if were going to two destinations (which I personally wouldn't, but many would) I would select two cities easily linked by (high-speed) rail. Also, if you choose more than one destination, be sure you do open jaw/muti-city/multiple destination tickets, so that you fly from the States to one city, and from the last city on your itinerary back to the States. Every time you change destinations, you spend time and money.
Ten days is just not a lot of time when you factor in jet lag, travel time, and travel hassles. I personally would want more than two days each in Rome, Amsterdam, and Barcelona. However, it's your trip, and your decision. Just make sure you are happy to spend that much vacation time on travel. If possible, I would add days to your trip, since usually getting to Europe is the hardest part, and you won't regret a longer vacation.
Hope you find a workable itinerary, and have a good trip!