It's been a while, so my DRG lore is a bit rusty.
1. Dragoons are specialized infantry of Ishgardian origin.
2. In general, it's implied that Dragoons are simply highly trained specialized infantry. The Azure Dragoon and some outliers (i.e. Heustinne) have draconic powers, but nothing concrete is given to explain Dragoons' abilities except "harnessing their inner dragon," or something like that.
3. Jump moves. That's about it. Player DRG abilities are derived from being an Azure Dragoon; what normal Dragoons can't do that the Azure can is unexplained, though it's fairly certain they couldn't use a move like Blood of the Dragon.
4. Like most everything else, an exact origin is not given. Haldrath is known to be the first of what would become known as Azure Dragoons, but beyond that no information is available.
... unless the Lore Book says more. Which it probably does. And my memory's not too foggy; I don't play DRG often.