EDIT: From Blizz's own article on the subject. "On Patch day, players will be able to launch the Battle.net desktop app and select the game you'd like to play--Burning Crusade Classic or WoW Classic. Both game types will display your characters that are currently playing with the old name/s of the realm. When you've decided on a game to play, choose the character you want to play and then confirm your selection. If you decide you'd like to play as your character in both game types it is possible to make use of an optional service for access to the cloned character for both games. All you have to do is login as you normally would and play."
Because the copy is complete prior to release Yes. The characters are all present in Classic, there is no downtime for copying in the case of Classic. If you choose TBC the copy may create downtime (just because its not instantaneous) and you can imagine a large number of characters being copied simultaneously and could cause the system to go down.