I'm looking for a data set with the names and INSEE (ID code) for all the French communes.
I am hoping to find a data set for the entire country. So far I have found that Wikipedia has it on the department level.
I think this might be what you are looking for (at insee.fr):
you can find the information you need in the Geofla® database. You can find it and explore it here on OpenDataSoft among with geolocation or population for every commune.
Disclaimer: I work at OpenDataSoft