You can make a func_door see-through too so there is no need for the func_brush. Set "FX amount" to something like 120 and it will be transparent.
You move the door by giving it an "open" input. The initiating "open" input can't come from the door itself, so create a logic_auto with the output "OnMultiNewRound - boing - open" (onMultiNewRound will be fired at the beginning of the round)
Alternativly, when the arrow isn't needed right from the start, make the door move later. So when a CP has to be captured first to get to the arrow, make the door start moving after the CP is caped. (OnCapture - boing - open). Reason is that the server doesn't need to send uneccessary informations about the door's position when it isn't needed.