The animation doesn't go off twice. You can just tell by the added damage and the TP return.
As for the accuracy, as the others have said, Jump accuracy seems to tie in with regular accuracy, even though there are pieces of equipment (i.e. Volan Greaves) that add pure Jump accuracy.
And as far as other jobs doing more damage on jumps, it is understandable. I know a MNK friend who would sub /DRG, boost up to max and then jump for great numbers. DRK using a scythe has a weapon with higher damage, so of course a jump would do more. And a WAR buffing up before with a Great Axe can.