For something to eat, how bout the best ice cream shop, not only in Paris, but very possibly anywhere: Berthillion on the Ile St-Louis. Eat in, or get a cone or cup at the window. Pay attention to days when they're closed, though.
For a special place, try the
Promenade Plantée (also known as the Coulée Verte). This is an old railway line that has been turned into a lovely place to walk and get some great views of Paris neighborhoods -- it's Paris's version of NYC's High Line.