This is funny, I was listing at your podcast this morning, and 1 hour after, I saw this article.
I live in Paris, so, I kind of know this town n_n
By the way, if you want to do something a bit different than touristy stuff, I know something amazing, that even a lot of people from Paris don't know : la promenade plantée.
This is really beautiful and different from everything you are going to see in this town.
This is a way, above roads, above people..., with beautiful nature.
I guess Google Images will be more helpful than my English abilities to help you to see what it is, haha !
If you want to take a nice walk and be able to see a part of Paris better than otherwise, this is for you.