I can see why someone would want to drive, rather than take a train, when travelling with young children - but like everybody else here I would advise (strongly!) against attempting this journey straight after an overnight flight. You say it's your first transatlantic flight - so you have no first-hand experience of what it does to your body. Listen to the frequent flyers here, who know what it's like. Don't tempt fate. You're going to be a danger to yourself and to others.

The other thing i was going to say is this... are you sure you really want to do this, with two children as young as three? You mention Disney, so I presume you're going to spend a few days there... but what other child-friendly activities are you planning? I can't think of much that one could do in Paris or Rome that would appeal to pre-school children. Wouldn't they get a lot more out of playing on a beach?