Just a suggestion--this is a lot to cover in one day. The Tuileries won't take as long as something like the Catacombes or Pere-Lachaise and are right across the river. You might want to consider doing those in the same day.
This is a great free place to go for an hour. You see some history of San Francisco, get to see the machines that make the system work, and after that you are in Chinatown, so can get some awesome food nearby.
Make sure to walk over to the Ile-St. Louis which is connected to the island where Notre Dame is by a footbridge. Less tourists, better place to grab a bite or ice cream (see Berthillon) and a cool neighborhood.