Any question answered...

The shared use path isn't controlled by the lights, so I'd say go for it.

I'd say so. I would also say it's a really bad design though, because as a pedestrian at the crossroads I have no indication that I should be expecting bikes, so I'd probably get pretty annoyed if someone cycled at me - only to learn there was a vague marking 50m down the road indicating that this is what you're allowed to do.

Different question, when you go onto the crossing on Street View, you can see a 'bike lane' sign. What exactly is this pointing at / signalling?

Here: https://www.google.com/maps/@51.4408928,-0.0640709,3a,75y,214.72h,86.93t/data=!3m6!1e1!3m4!1sZ8UZMDONyiBJPSjNsM5JHQ!2e0!7i16384!8i8192

Does the 'bike lane on the pavement' marking on the left mean that bikes can hop onto the pavement and ignore the light if turning left?

Seen others do it but assumed they were RLJing.
Realised I don't know what that type of road marking means.

The marking is what is known as a 'jug handle' (because of its shape) and its purpose is to introduce shared-use provision (walking and cycling) on the footway there to enable cyclists to ride up to the 'Toucan' ('two can' cross, i.e. people walking and cycling) crossing, which they obviously wouldn't be able to access unless they were on the shared-use 'footway' first. If you look at the whole junction, all of the crossings have the dual provision of the normal pedestrian lights and the additional cycle green light. This means that all the footways around have to be shared use, and you can indeed see more 'jug handles' and the blue 'shared use' diagram up on the lamppost on the footway around that left turn.

In one of the many little absurdities usually thrown up by these crap designs, there's no 'jug handle' where the dropped kerb is for the residential driveway entrance, probably because the distance between the corner and the driveway entrance is too short for one and it seemed unnecessary to the engineer. By the logic of the design, there should really be one, both for clarity and because there doesn't seem to be any indication of where shared use really ends. I can't see any indication that shared use continues along the A205 London Road, but you could perhaps argue that the signing and marking is misleading if you were stopped for riding along the footway.

But yes, the signs indicate that shared use is lawful along this piece of footway around the junction and you're quite within your rights to mount the footway and turn left in this way. You should obviously take extra care in re-entering the carriageway at the driveway, because chances are that when you turned the corner the main drag will have had a green light, or that Sydenham Hill will have had a green light in the meantime, while you have to look up to 180 degrees behind to see what's going on. Both factors increase the possibility that you may come into conflict with traffic on the A205 after turning left at red in this way.

Pretty sure that the pavement bike lane is there for the benefit of those cyclists who want to turn right (all other traffic 'has' to turn left) using the controlled crossing, and then cross the the South Circular (using a second controlled crossing) to join the eastbound side.
Or am I stating the bleedin' obvs?

ETA - Beaten to it by O.Schick.