Private.
I've been told there were two Sith at the time. Palpatine and his apprentice, Darth Vader, a talented and powerful Jedi that Palpatine managed to corrupt and manipulate. At some point in history the Sith destroyed themselves from within and created a new rule about how there would only be two at a time, but let the rest of the world believe them to be extinct. A foolish waste of history and knowledge lost in the process. They apparently remained well hidden until Palpatine's machinations came to fruition, as he hid his true nature from everyone. He was quite good at it apparently, as no one discovered it until it was too late.
I don't think it's as simple as the New Republic "allowing" the First Order to be created. I only arrived here a few months ago and haven't had the opportunity to speak to those in charge, but nothing in politics is ever that simple, especially after a war. Though I highly doubt a Jedi led government would have fared much better.
There was. It was ruled by a power hungry Sith known as Emperor Vitiate. He was so mad with power that we joined forces with the Sith after the Galactic Civil War to try and stop him.