The idea started with the invention of the piano and the possibility of sustaining sound with the damper pedal (see the site I suggest below for a full story).
Yes, indeed. Paul Wittgenstein lost his right arm on World War II and commissioned several composers to write pieces for him (the most famous of those being Ravel’s left hand concerto). Incidentally, Paul was brother to Ludwig – the famous philosopher.
Another famous pianist who lost the use of his right hand due to practice induced injuries is Leon Fleisher who also has had many pieces written for him (I hear that after almost thirty years he recovered some of the movements in his right hand and is now starting to give recitals again with the conventional repertory)
There are far more pieces written for the left hand alone, but a few for the right hand also exist (see the site below for a most intriguing explanation why most of the repertory is for left hand).
The most complete reference I came across for one hand repertory is:
Theodore Edel – Piano Music for one hand (Indiana University Press)
And this site lists most if not all of the repertory: (all your questions will be answered!

)
https://hjem.get2net.dk/Brofeldt/And Scarlatti never wrote any piece for left hand alone. It was Scriabin (and as ayahav said it was not a concerto).
Best wishes,
Bernhard.