Is the back firing arc of the Firespray a real firing arc? I mean, does it count for the cards that say "inside/outside the ship firing arc"?

Yes. "Arc" ruleswise refers to the printed arc on the ship's base card.

As mentioned that Auxiliary Arc is still an "arc" as far as things like Backstabber and Outmaneuver are concerned.

Try NOT to confuse it with most turret weapons. Those ships still have a firing arc but allow targeting "outside of your firing arc" to get the 360 degree effect. I guess I'll also mention that a ship with no firing arc (Rebel Transport) means you can NOT be inside OR outside of its firing arc as it doesn't have one.