The stream is 480p to let some place on the bandwidth for all the commands. Bebop records video in 1080p, video you can download and watch after the flight.
Right now, you connect your device to the Bebop by Wifi and only one device can be linked to the Bebop. So, to me, you can't send the stream to somewhere else. I think it is planned to add a 4G chip on the Bebop which will allow you to connect the Bebop to your device in Wifi and to the Cloud by 4G as the same time. Someone from Parrot can backup or correct this information