I posted in your question about apartments/hotels in the review area, but will repeat here in case others see this too.
We went in July with our daughter - she was 5 but turned 6 just after we got home. She loved riding the metro (her favorite part of the whole trip), going to Versailles (the gardens and Hall of Mirrors were the highlights), the Eiffel tower - especially watching it sparkle from the Trocadero at 10pm, the Catacombs (def a big hit, make sure you know the release schedule to get tickets), and the playground in the Jardin du Luxembourg (Ludo Jardin; worth the few euros entry cost). She also enjoyed parts of the Louvre- the Sphinx, the jewels in the Apollo Gallery, and some of the other bigger sculptures.
We did a lot of museums because the older cousins (11 & 15) wanted to see everything and it was their first time - Louvre, Orsay, Cluny, Les Invalides. So we loaded books on our kindle and our daughter read a lot in the museums - but it worked perfectly because the adults/older kids got to see what they wanted and she was happy reading on a bench until we went to the next room. We might have made different choices of course if it was just us with her, like you will be.
Will also say - generally assume you can bring in water and food, as long as you don't eat in areas it's prohibited. A lot of times we didn't bring it with us as it was technically against the rules. But I didn't see anyone get stopped at any of the sites we visited, the bag checks were all about safety/weapons. So we bought lunch at several places, but could have just eaten food we brought for a quicker/easier experience.
-beth