It sounds like you will need to strip some of the insulation off the 'hot' wires as all the red solder points in the wiring diagram need to be made onto bare wire. This sounds like it may also entail stripping back some insulation on the overall cable insulation so you can expose some more of the earth/ground/shield wire (which is woven around the single insulated signal/hot wire), pull back the shield wire so that you have more of the signal wire exposed in order to get a longer length of the wire to strip the insulation back from.
I don't know what tools you have handy, but a pair of small wire cutters is always very handy when doing any soldering work, something like this:
Not too big a pair; ones around 100-120mm in length will do. You can get much bigger cutters with a similar shape but they are more for cutting domestic mains (or larger) wiring and generally don't have fine enough blades for small work and they won't fit into small spaces.
You can also buy dedicated wire stripper tools, which are a good investment if you do a lot of soldering. I have some like this:
You adjust the screw so that you've got an opening in the jaws that's the approximate size of the central wire core, clamp the jaws round the wire and pull off the insulation. Good for single wires, less good for stripping the outer insulation off complete cables with a shield layer and a central core.
You should be able to find very low cost versions of the above at places like Jaycar or Bunnings. For serious amounts of work you can easily pay $200 for some wire cutters that will stay really sharp for ages, but for under $10 you'll get some that will be sharp enough to start with but will loose their edge over time.
For the outer insulation on the pickup wire, I'd probably run a sharp blade around the insulation first, being careful not to press too hard. You should be able to feel the blade when it runs across the shield wire. Don't press to hard as the shield cable strands are thin, and whilst you can cut though a few without any ill-effect, you don't want to cut through too many. Once you have scored through the insulation, then put the wire cutters on the wire at the score line, gently squeeze them a bit so that the blades catch on the insulation then (without increasing the pressure on the blades) pull the cutters towards the end of the wire so that they pull the insulation off.
You can then pull the shield wiring back to the new insulation cut point and twist the shield wiring so that you get a more solid piece of wire.
With the central wire, I'd either use dedicated wire strippers to expose more wire, or use the wire cutters to strip the insulation off, using just enough pressure to bite into the insulation with enough grip to pull it off, but without cutting through the wire itself.
If you go and buy some cutters and/or strippers, I'd buy a small coil of thin wire at the same time to practice on. You don't want to make mistakes on the real thing if you can avoid it.
As for the second question, If you look at the PBG wiring diagram, I'd do it like that and solder both pickup wire shields/grounds/earths to the back of the volume pot. The back of both pots are linked to ground, so the reality is that it doesn't matter which pot they get connected to, so it's best to do what's easiest from a physical layout point of view.
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