They will fly Rafale Ms and there are several on-board the Charles de Gaulle already trained for a 'Falklands scenario'. Equally some French pilots have been trained to fly Typhoons.
https://www.gov.uk/government/news/double-first-for-french-and-british-fast-jet-pilots
They are F-35B STOVLs.
I'm not sure whether the French Carrier sharing is only a stop gap until the QEs come out. There is a close co-operation between French and British forces right now, so whether there is an F-35B/Rafale M carrier sharing future or not I don't know.
They only have a ramp capable carrier - the Cavour 550, so my guess would be F-35Bs for their navy too. They also plan for F-35As to replace AMXs and Tornados.
http://www.defensenews.com/article/20120515/DEFREG01/305150010/Italian-AF-Navy-Head-F-35B-Showdown
STOVL was always going to complicate things. God knows why the F-35A couldn't simply have been a Raptor with bigger internal bays. Maybe a Strike Raptor is in the future somewhere.