The Redax ignition and alternator kit is an excellent upgrade, but I understand that you just wanna get the bike running to assess the condition of the engine and transmission before you go spending money on upgrades. You should be able to get it fired up for very little cost. If you can get the pickups sorted, all you really need to get it running is a pair of coils.
The good news is that you can resurrect those pickups if there's enough wire still attached to them. Remove the old insulation with as little damage to the actual wires as possible, then clean the ends of the bare wires so they'll take solder. Solder new wires onto the existing bare wire tails, then slide heat shrink over them to cover the bare sections. Push the heat shrink tube hard up against the pickups and shrink it into place. Seal around the base of the heat shrink at the pickup end with heat resistant mastic and re-clip the wires using the metal tag retainers on the pickups. Job done. Note that they're polarity sensitive when you wire them to the DMC1 unit. I think the installation instructions for the DMC identifies the pickup polarity by wire colour. But it looks like the original wires on yours have been replaced by a previous owner, so you'll probably have to resort to trial and error. If you don't get a spark when you crank the engine, try swapping the pickup leads around.
If the pickup wires are too far gone to recover, your best option would be to scrounge another pickup plate in better condition. Guys like Red or Keith Nairn might have a bunch of old pickup plates lying around that they've removed from customer bikes when fitting upgraded ignition/alternator kits. With a bit of luck it'll come with another alternator coil attached.
Rob is absolutely correct about the alternator coils. The coils don't earth through the mounting bolts. There should be a yellow wire sprouting out of one end and a yellow/black wire at the other end. Simplest way to wire the two coils is to connect them in parallel so they work as a single phase. Join the yellow wires from both coils together, and the yellow/black wires together, then treat them as you would a normal single phase alternator.
And to add to your confusion, the upgraded coils can be run without a regulator.
Not advisable. With two alternator coils your battery will have an easier and longer life with a voltage regulator. Pretty much any RR will do, but if you decide to stick with the setup long term, I'd suggest a single phase RR as used on a bunch of Ducati models such as the MHR or Pantah (there are aftermarket versions available). The advantage of that unit is that it has an output for a dashboard charge light. There's a "GEN" light in the tacho (normally disconnected) that you can make use of.