How do I deal with very small wire which is too narrow to fit properly into a breadboard?
Should I solder them to male jumpers


Yes.


  Use a combination of heat-shrink tubing and 1/8 inch electrical tape for insulation?


No, the size is too big for proper heat shrink (unless you have one sufficiently wide) or tape to help much. For the most part, you can use hot-melt glue sticks, or epoxy to insulate and help support it. Frankly though, once you solder the wires, you really don't need to insulate it, unless you'll have other wires loose. If you are careful or don't unplug and plug it in a lot, it will last a long time.