In the first video, a US Navy C-2A Greyhound of VRC-30 Providers makes a trap and catapult launch from the French carrier Charles de Gaulle. French pilots also train train to operate from American carriers.
The C-2 is a derivative of the E-2 Hawkeye radar plane, with the radar dome and other equipment removed. It is used for replenishing supplies for the carrier battle group, called Carrier Onoard Delivery (COD).
In this video, F/A-18 Hornets perform a series of touch-and-go landing, cable traps, and launches from Charles de Gaulle.