we have traveled extensively with our sons, now 11 and 14, since the time they were babies. they now think the lax-london flight is "short".

our favorite kid-friendly airline is air new zealand, but i doubt they fly from sf to rome. virgin is also fine; even babies love the movies and video games.

best advice is to take night flights and take along her carseat. if she does not sleep, take turns walking around and reading with her. we each took 1 or 2 hour shifts.

we did need to "drug" our younger son occasionally with benedryl (at our pediatrician's suggestion). worked well, and so far he seems unaffected. but if you plan to use benedryl, test it before you leave, because it makes some kids agitated.

Play-doh on the plane? Wouldnt do that; too messy.
We've always flown with our children, even when they were very young, and never had a real problem. Do pack enough for on the plane; a change of clothes in case she's sick, and plenty of food. Recently we were on a flight to the US that took twice as long as planned; all the time we were on the plane, and they had long since run out of food. Luckily our children are older now, but I felt very sorry for those with small children. I will always carry some emergency chocolate bars or similar in the future!

I'd recommend "Staying Home with Grandparents Airlines". The entire trip, not just the flight, is much more enjoyable for all concerned.

Mebe,
If your daughter enjoys playdough, try Crayola's model magic. It is like playdough, but not messy. It comes in small indiv. packets. We just pack a ziploc to put it in after play. If you leave Model Magic to air dry it hardens. It keeps kids busy and is not messy.
Have a great trip.