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With a YT-1300, Decimator, or YT-2400, Tactician says "If you make... inside your firing arc." Does this mean that one of these ships can use it against anything thy shoot, since their "arc" is 360 degrees?

No. Ships with a turret still have an arc printed on their base, and use it whenever a rule checks if a ship is within their arc (shooting secondary weapons, outmaneuver, etc). The only thing the turret primary weapon does is allow the ship to make attacks outside of its arc. So, since tactician requires the target to be within the firing ship's arc, it can only be used in the ~90* forward arc and does not apply to out-of-arc shots.

(Note that since the Firespray has two primary weapon arcs tactician works in both of them.)

No, it doesn't. It has a primary weapon arc and an 'auxiliary arc' - so pointedly, for example, you can't fire a heavy laser cannon out of your backside.

But the rest is correct. Whilst turreted ships can fire 360', their 'arc' still exists and is relevant for anything which requires you to be in arc (like tactician or secondary weapons) or opponents to be out of arc (like outmanouvre or Backstabber's pilot ability)