a) How does the Phantom appear in the mirror?
It is the same effect as a 2 way mirror, actually any piece of glass/perspex that is tinted with a mirror/reflective tint, once light is cast behind it, you will see the person behind the mirror appear. In Vegas, there was a hole in the mirror and as the mirror slid accross, Christine would walk through the hole making it appear (with added smoke) that she walked through a mirror that didnt open
b) How do the phantom and Christine get to the catwalk so quickly at the beginning of the Title Song?
Most of the song is pre-recorded, so as the Phantom and Christine are getting out of the mirror setup, a set of doubles cross the stage and go down the trapdoor, another set of doubles is ready up on the bridge and start to cross. On Broadway (sometimes, not sure if all the time) the real Phantom and Christine are also on the bridge and swap with the doubles. If not, then the real phantom and christne come out in the boat, also sometimes the verse when they are in the boat is live singing from there.
c) After the reprise of Wishing, how is the sound of Raoul and Christine singing heard if they are offstage?
when raoul and christine exit through the trapdoor, there is a monitor of the conductor right under the stage, they watch him to keep time and sing from there.
d) How does the phantom disappear at the end?
, that was quite well explained above, except i think for the knees up by the chest bit, last time I saw it done the whole chair just sort of slides back so if you sat on the throne you basically would be sitting on his lap.
e) How does the boat move?
remote/radio control
f) How does Raoul jump off the bridge yet stay safe after Madame Giry leads him towards the Phantom's lair?
the trap door is lined with lights so raoul can clearly see it. Inside is a big piece of foam, he lands and goes off to get wetted down, then pops back up through the same hole. Back when the show toured, there was no trap door so he jumped onto a dark air cushion or sometimes they just cut that out and he runs off the bridge in the same direction as the rat catcher
http://www.flickr.com/photos/27199361@N08/ Phantom at the Royal Empire Theatre