Some of my suggestions:
If you are in France during the Summer Soltice, make it a point to be in Paris for the La Fźte de la Musique:
http://gutsytraveler.com/paris-free-music-festival-on-the-summer-solstice/There are bands on every corner, the entire city is awake late into the night, and there is something great about hearing 24 year old Parisian street band singers sing affected covers of The Doors and other classics. (side note, Jim Morrison's grave is in Paris, and the entire cemetery he is buried in is a testament to French sculpture).
The best meal I've ever eaten was the Fixe Prix option at Procope:
http://www.procope.com/Get the escargot, veal, and tiramisu. Whatever you do do not let yourself get talked out of getting the escargot. Think shrimp bathing in butter, then seasoned with butter, and then dipped in butter. In fact, for every meal, get extra butter. They do butter better than wine in France.
Depending on your affection for history, I would say that as an American in France you owe it to yourself and daughter to go to Normandy. There are copious options to take trains and buses, and it is well worth it. The actual beaches are remarkable, but the cemetery is... hard to put into words.
In Paris, the worst thing you can do is force the sites. Don't try and do the full city in three days or you'll exhaust yourself and hate the experience. You'll enjoy the city much more if you take a picnic up to the hill on the side of Sacre Couer, sit at a café for an hour near the Pompidou, or explore Montemarte around the wall of love (Mur des Je taime) for a few hours.
You'll be in Europe during the Euro Cup (like the World Cup but just for Europe). Find our when France is playing England and go to the biggest viewing party you can (depending on the maturity of your daughter--they're not much better than College Football Fans) and soak in the experience. If they make it deep then things get really hectic and they'll do things like show matches at Notre Dame and other large public spaces.
A train ride down to Cannes or Cannet will introduce you to some wonderful view of rural France, and the seafood along Cannes' boardwalk is remarkable. Cannet has fragrances houses, which is fun if a bit touristy.
Parisians are like any other big city people-if you try and talk on the train or while they're getting things done, you'll be told they don't speak English to avoid you (they do). But if you perfect your "I'm a tourist and need help and am sorry to bother you" face you'll find most everyone is remarkable nice.
France is great. Get lost in it with a pocket full of Euros to keep hydrated with crepes and ice creams and fantas from the sidewalk vendors you'll pass by every block.