I'll help with what I can, but I'm sure someone else would be able to add or correct any mistakes I make. 
1: As far as known, yes, Dragoons are only found in Ishgard.
2. There are actually two different kinds of DRG, just like there are two different kinds of BRD (the song-singing entertainer kind and the ARC-based bowmen and women who sing to buff their compatriots in battle). The first kind of DRG are basically Ishgard's version of LNCs, who are specially trained to fight dragons and so adopt various techniques for dealing with their often aerial Nemesis. Accordingly Ishgard has an actual military order of knights consisting soley of DRGs - fittingly called the 'Order of the Knights Dragoon'. The second type of DRG is what the player actually is, the so-called 'Azure Dragoon'. Using the Eye of Nidhogg to grant them extraordinary stamina, strength and jumping potency even over their lesser kin (see next question), the Azure DRG served as pretty much Isghard's 'hero'. Usually there is only one Azure DRG at a time... that is until the player came along and the Eye anointed them as a second Azure DRG concurrent with the serving one (Estinien).
3. A 'normal' DRG and an Azure DRG seem to posses the same sort of abilities, in that they use regular lance skills and specialised Jumping attacks (that allow them to scale great heights and leap onto their flying adversaries). The Azure DRG however, also makes use of the immense wellspring of aether that is the Eye of Nidhogg (which is the source of the player's DRG Soul Crystal), actually drawing on the draconic power that flows through the Eye to enhance their power imensely. Additionally, the DRG Artifact Armour was said to have been specially forged in dragon blood, further enhancing the draconic power they use. Unfortunately, this naturally swings both ways, in that it also allows the dragons to negatively influence the Azure DRG in turn if their mental state is not strong enough, especially Nidhogg. Estinien struggles to resist Nidhogg's influence throughout the level 30-50 DRG Job quests and Heavensward's storyline,
culminating with the 3.0 story's ending scene where, jubilant in Thordan's defeat and possessing both Eyes of Nidhogg (and thus doubling Nidhogg's influence), he lets his mental guard down momentarily allowing Nidhogg to seize control of him and use him as a host.
4. The DRG Job seems to have it's origins in the very founding of Ishgard, when
Thordan the 1st and the Knights Twelve betrayed the dragons, his son and legal heir to the throne Prince Haldrath, who seemed to already be a DRG, however refused to ascend the throne and instead "bear penance" for his part in Rattatoskr's murder as "a hellsbound hunter of dragons" - in the Echo scene the player sees when meeting Hraesvelgar for the second time Haldrath is clearly seen wearing DRG AF and bearing Gae Bolg, the DRG relic weapon, perhaps suggesting that the Job actually predated Ishgard.
But one of the Loremasters would be able to clarify this further.
Hopefully this has helped.