All servers currently in use are being converted to TBC servers from FryeJacob's blog

All servers currently in use are being converted to TBC servers. New Vanilla versions of these servers will become available. You get the choice of activating your character on the TBC server or WOW Classic TBC Gold the Vanilla version. If you wish to activate the same character on both, you have to pay.

Edit: Blizz's own piece about it. On the day of patching, you'll be able to access the Battle.net desktop and choose the game you would like to play: WoW Classic or Burning Crusade Classic. Both games will show your characters that are currently playing under the old realm name/s. Once you've chosen the game you want to play, you'll select the character you want to play and confirm your selection. If you decide you'd like to play your character in both types of games it is possible to use an optional paid service to access the cloned character for both games. Then all you need to do is sign in as you normally would and play."

Yes, since the copying is completed prior to buy TBC Classic Gold release. All the characters exist in Classic, there is no copy downtime in the case of Classic. TBC could cause downtime, due to the fact that it isn't instantaneous. It can also result in a lot more characters being copied at the same time. This could lead to the system being unable to function properly.