PVC threaded fitting broken off. How to remove

You can use the Sharkbite or Gatorbite connectors as a junction for copper and PVC. Sharkbite and Gatorbite connectors are "snap together" connectors available at Lowes or HomeDepot, you can put copper in one side and PVC in another.

I would think PVC piping would be fine. You can even paint it any color you like.

Alternatively, you could attach the torches themselves to the fence, but with the piping, you can remove the torches in Winter.

I would think cable ties would be fine, or clamps as shirlock suggests. IMO the trick would be to make sure the pipes don't slide up and down and come out of the fasteners. You could drill a hole in the pipe segments, and cable tie that to a horizontal piece of the fence to combat this.

Most hardware stores will have a variety of PCV piping. If the PVC pipe is thing enough (which it should be if these are not huge tiki torches), you can get a special cutter for them, which is sort of a curved pair of scissors. Makes it a piece of cake to get straight cuts. If it is thicker, any saw will do, as PVC is generally very soft.