If I use a Beolink Passive with a pair of non B&O passive speakers connected through a subwoofer in a link room, does the Bolink passive unit turn the speakers on and off from the Beo4 and off automatically after the music finishes?

Yes, though it probably won't turn the subwoofer on/off (if it is powered) unless it is able to sense the signal coming through when you use the speakers.

The passive should turn off if the main system goes into standby which is the same if you were using B&O active speakers on a Beolink system. This is assuming that the source is CD. If you are using the radio it will stay on until you put either the passive or the whole system into standby.

Peter