Tagron & Jan,
I am pretty sure that this is joined in France. "Le service des douanes" in France, is both import / export, customs and immigration - I have never seen any separation between the two services. Nor do I believe there is another word to describe these individually. When you request customs at an airports that may not always have customs based there, you will be sending a request in writing to an address @douanes.gouv.fr (or douanes.finances.gouv.fr) - not any other service. When you are at an airport in France, it is the "douaniers" that check your passport. Some may be wearing a police like uniform, but they should also be wearing the insignia (badge).
Some police forces have been helping the douaniers, depending on the local circumstances which may be why there is some confusion.
Re: Cherbourg - contact them and ask them how much notice they require, in some places in France, occasionally they require no notice at all because they already have the "douaniers" there the whole day for other reasons, and sometimes they require it at least 24 hours in advance (or longer if weekend).
carlmeek - this is excellent work! I really appreciate it!