1, Who makes them
A part of the Jedi training is to construct his or her own lightsaber, which they can pretty much do wherever. Luke created his second lightsaber by himself, for instance.
As for the Sith, they create their own lightsabers as well. To be fair, in the EU it is not unheard of for Sith Masters to reward their apprentices with lightsabers they have not created themselves, for example
Darth Bane's curve-hilted lightsaber was given to him by his by one of his instructors at the Sith Academy he was attending.
But of course, those lightsabers are also created one way or the other, and in that case
It was created by the instructor's master.
So, to sum that up:
The Jedi and Sith create their own lightsabers, there is no 'lighsaber manufactor'
To address the point that they are rare in the original movies and episode VII. It is not really that strange that there aren't many lightsabers around when there aren't many Jedi or Sith around. In the prequels there is no shortage of lightsabers.
2, How do they work
The lightsaber blade is a plasma beam, and it is powered by a crystal. The crystal is also what determains the color of the blade. As Hypnosifl pointed out, the crystal is a kyber crystal. They could be found in the ice caves of Ilum, and a few other places. Apparently the kyber crystals are attuned to the Force and lacked color before being chosen by a Jedi. Once chosen, most crystals turned blue or green.
There are also synthetic crystals. These are artificially created crystals that are often much more powerful than the naturally formed ones. The Synth-crystals have a fixed color, most often red, and are used regularly buy the Sith. However, Luke's second lightsaber is powered by a green Synth-crystal.
3, Psychic phenomena
Unless you mean The Force, I don't know what you mean by that. The Force is what allow some individuals to get extraordinary powers, and my age has nothing to do with it! Oh, I mean the lightsabers has nothing to do with it.