We see later on when Suzaku is remembering what he did when Lelouch ordered him to live what happened, Suzaku closed the cockpit of the Lancelot with Lelouch with him and went to escape, while i can't remember it entirely, he probably picked up Euphie on the way while Kallen chased after him knowing Zero was inside.
Because the Hadron Cannon was unfocused, C.C must have used the confusion to do something to the ruins remotely, possibly using the Thought Elevator, to send them to the island, it might have been area of effect as she probably was only targeting Lelouch but Suzaku, Euphie and Kallen was within the vicinity. C.C probably had planned that Lelouch would learn more of Geass, of Charles and V.V and of the Thought Elevator so that Lelouch would be prepared to grant her wish in the end.
How the Black Knight got out of the Avalon's attack, well as i said the Hadron Cannon was unfocused, it was still a prototype and Lloyd expresses how annoyed he is later on when Rakshata is able to perfect it by focusing the beams, because of the way the Hadron Cannon attacked and what we see it's assumed that the Black Knights retreated.
As for how they couldn't get in contact with Lelouch or Kallen, i'm quite sure what ever C.C did separated Suzaku and Kallen from their Knightmares and obviously they at the time were relying on their comms within the Knightmares so didn't carry radios, this is proven by the fact that the playback of Suzaku disobeying order was recorded in the Lancelot, the Britannian forces probably weren't aware of Suzaku or Euphie being on the island as they where there by chance on a separate mission investigating the ruins which Bartley suspects was the reason why Japan (and a number of other nations) was invaded in the first place.