At a minimum what would I need to repair a fracture in a gold pendant?
In my jewelry making course way back in 2000, heating metal for soldering required actual flame... a torch. You wouldn't want to do this with the equipment you have at all. It's a totally different process.

There are two different soldering methods in jewelry making: soft solder and hard solder. While many craft-y projects can be assembled using soft solder (that is, solder that can be melted using a soldering iron), working with silver and gold requires hard solder and a torch. There's a guide to it here.


  "Hard" solders are alloys that melt at slightly lower temperatures than the corresponding metal pieces to be fused. Silver and gold jewelry require hard soldering. Copper, brass, and bronze components can also be fused with hard solders. In addition to requiring flux, hard solders generally also require the use of pickling solutions to clean and remove surface oxidation from the finished piece. 


This matches my experience pretty much exactly. While we largely worked with base metals - copper, brass, nickel silver, etc - we did all of our soldering using a torch setup, flux, and pickling solution.


  Metal workers typically use oxy/propane or oxy/acetylene torches with hard solder, but many soldering tasks can also be achieved with butane torches. Use a butane torch to solder jump rings, create ball-end head pins, make custom wire links and chain, create small bezels, fuse together small metal components, etc. 


For your little project, based on this section, I'd guess that you could get away with using a butane torch but you'd also need to buy gold solder (to match the color), flux, pickling solution... which was always kept warm in an old Crock Pot.